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Sparsity-certifying Graph Decompositions

We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the $(k,\ell)$-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.

Sparsity-certifying Graph Decompositions Ileana Streinu1∗, Louis Theran2 1 Department of Computer Science, Smith College, Northampton, MA. e-mail: streinu@cs.smith.edu 2 Department of Computer Science, University of Massachusetts Amherst. e-mail: theran@cs.umass.edu Abstract. We describe a new algorithm, the (k, `)-pebble game with colors, and use it to obtain a charac- terization of the family of (k, `)-sparse graphs and algorithmic solutions to a family of problems concern- ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri- zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k, `)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. 1. Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems

Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems

Descomposiciones del gráfico de certificación de la sparsity Ileana Streinu1*, Louis Theran2 1 Departamento de Ciencias de la Computación, Smith College, Northampton, MA. Correo electrónico: streinu@cs.smith.edu 2 Departamento de Ciencias de la Computación, Universidad de Massachusetts Amherst. Correo electrónico: theran@cs.umass.edu Resumen. Describimos un nuevo algoritmo, el (k, `)-pebble juego con colores, y usarlo para obtener un charac- la terización de la familia de gráficos (k, `)-sparse y soluciones algorítmicas a una familia de problemas ing árbol descomposicións de gráficos. Casos especiales de gráficos escasos aparecen en la teoría de la rigidez y tienen ha recibido una mayor atención en los últimos años. En particular, nuestros guijarros de colores generalizan y fortalecen los resultados anteriores de Lee y Streinu [12] y dar una nueva prueba de la Tutte-Nash-Williams carácteri- Zación de arboricidad. También presentamos una nueva descomposición que certifica la esparcidad basada en la (k, `)-pebble juego con colores. Nuestro trabajo también expone conexiones entre los algoritmos de juego de guijarros y anteriores algoritmos gráficos escasos de Gabow [5], Gabow y Westermann [6] y Hendrickson [9]. 1. Introducción y preliminares El foco de este documento son las descomposicións de (k, `)-sparse gráficos en bordes-disjunto subgraphs que certifique la escasez. Usamos el gráfico para significar un múltiplo, posiblemente con bucles. Nosotros decimos que un grafo es (k, `)-sparse si ningún subconjunto de n′ vértices abarca más de kn ` bordes en el gráfico; a (k, `)-sparse gráfico con kn ` bordes es (k, `)-estrechado. Llamamos al rango k ≤ 2k−1 el superior rango de gráficos escasos y 0≤ k el rango inferior. En este artículo, presentamos algoritmos eficientes para encontrar descomposicións que certifiquen la escasez en el rango superior de `. Nuestros algoritmos también se aplican en el rango inferior, que ya era ad- vestido por [3, 4, 5, 6, 19]. Una descomposición certifica la escasez de un gráfico si los gráficos dispersos y los gráficos que admiten la descomposición coinciden. Nuestros algoritmos se basan en una nueva caracterización de gráficos escasos, que llamamos el juego de guijarros con colores. El juego de guijarros con colores es una regla de construcción de gráficos simples que produce un gráfico escaso junto con una descomposición certificadora de la escasez. Definimos y estudiamos una clase canónica de construcciones de juego de guijarros, que corresponden a previamente estudiado las descomposiciones de los gráficos escasos en los árboles disjuntos del borde. Nuestros resultados proporcionan un marco unificador para todos los casos especiales conocidos anteriormente, incluidos Nash-Williams- Tutte y [7, 24]. De hecho, en el rango inferior, las construcciones canónicas de juego de guijarros capturan la propiedades de las rutas de aumento utilizadas en los algoritmos de unión de matroides y de intersección[5, 6]. Dado que los gráficos escasos en el rango superior no se sabe que son uniones o intersecciones de la matroides para los que hay algoritmos de ruta de aumento eficiente, estos no se aplican fácilmente en * Investigación de ambos autores financiada por la NSF bajo subvenciones NSF CCF-0430990 y NSF-DARPA CARGO CCR-0310661 al primer autor. 2 Ileana Streinu, Louis Theran Significado del término Gráfico escaso G Cada subgrafo no vacío en n′ vértices tiene ≤ kn ` bordes El gráfico ajustado G G = (V,E) es escaso y V = n, E= kn− ` El bloque H en G G es escaso, y H es un subgrafo apretado El componente H de G G es escaso y H es un bloqueo máximo Gráfico cartográfico que admite una orientación de grado-exactamente-uno (k, `)-maps-and-trees Edge-disjunt union de ` árboles y (k- `) map-grpahs `Tk Unión de ` árboles, cada vértice está exactamente en k de ellos Conjunto de piezas arbóreas de un `Tk inducido en V ′ ́V Piezas de árboles en el `Tk extendido por E(V ′) `Tk Apropiado Cada V ′ V contiene ≥ ` pedazos de árboles de la `Tk Cuadro 1 Gráfico escaso y terminología de descomposición utilizada en este artículo. el rango superior. Pebble juego con construcciones de colores por lo tanto puede ser considerado un fortalecimiento de caminos de aumento a la gama superior de gráficos de la escasez matroidal. 1.1. Gráficos escasos Un gráfico es (k, `)-sparse si para cualquier subgrafo no vacío con bordes m′ y n′ vértices, m′ ≤ kn `. Observamos que esta condición implica que 0 ≤ ` ≤ 2k− 1, y a partir de ahora en este Haremos esta suposición. Un gráfico escaso que tiene n vértices y exactamente bordes kn se llama apretado. Para un gráfico G = (V,E), y V ′ V, utilizamos el intervalo de notación (V ′) para el número de bordes en el subgráfico inducido por V ′. En un gráfico dirigido, out(V ′) es el número de bordes con la cola en V ′ y la cabeza en V −V ′; para un subgráfico inducido por V ′, llamamos a tal borde un borde superior. Hay dos tipos importantes de subgrafías de gráficos escasos. Un bloque es un subgrafo apretado de un gráfico escaso. Un componente es un bloque máximo. La Tabla 1 resume la escasa terminología gráfica utilizada en este artículo. 1.2. Descomposiciónes de certificación de la sparsidad Un k-arborescencia es un gráfico que admite una descomposición en k borde-desjunto que abarca los árboles. La Figura 1(a) muestra un ejemplo de una 3-arborescencia. Se describen los gráficos k-arborescentes por los conocidos teoremas de Tutte [23] y Nash-Williams [17] como exactamente el (k,k) apretado gráficos. Un map-graph es un gráfico que admite una orientación tal que el grado de cada vértice es Exactamente uno. Un k-map-graph es un gráfico que admite una descomposición en k borde-disjunto mapa- gráficos. La Figura 1(b) muestra un ejemplo de un 2-map-graphs; los bordes están orientados en uno posible configuración que certifica que cada color forma un mapa gráfico. Los mapas pueden ser equivalentes definido (véase, por ejemplo, [18]) como tener exactamente un ciclo por componente conectado.1 A (k, `)-maps-and-trees es un gráfico que admite una descomposición en k− ` borde-disjunta - mapas y árboles que se extienden por los árboles. Otra caracterización de los mapas, que utilizaremos ampliamente en este artículo, es la siguiente: los gráficos (1,0) ajustados [8, 24]. Los k-map-graphs son evidentemente (k,0)-stight, y [8, 24] muestran que lo contrario se sostiene también. 1 Nuestra terminología sigue a Lovász en [16]. En la literatura matroide los mapas a veces se conocen como bases del matroide de la bicicleta o pseudobosques que se extienden. Descomposiciones del gráfico de certificación de la Sparsity 3 Fig. 1. Ejemplos de descomposiciones certificadoras de la escasez: a) una 3-arborescencia; b) una 2-map-graph; c) una (2,1)-maps-y-árboles. Los bordes con el mismo estilo de línea pertenecen al mismo subgrafo. El 2-map-graph es se muestra con una orientación certificadora. Un `Tk es una descomposición en `árboles disjuntos de borde (que no necesariamente abarcan) de tal manera que cada uno vértice está en exactamente k de ellos. La figura 2 a) muestra un ejemplo de un 3T2. Dado un subgrafo G′ de un gráfico `Tk G, el conjunto de piezas arbóreas en G′ es la colección del componentes de los árboles en G inducidos por G′ (dado que G′ es un subgrafo cada árbol puede contribuir piezas múltiples en el conjunto de piezas de árbol en G′). Observamos que estas piezas de árboles pueden venir del mismo árbol o ser un solo vertex “árboles vacíos.” También es útil tener en cuenta que la definición de un árbol-pieza es relativo a un subgrafo específico. Una descomposición `Tk es apropiada si el conjunto de las piezas arbóreas de cualquier subpárrafo G′ tienen un tamaño mínimo `. La Figura 2(a) muestra un gráfico con una descomposición 3T2; observamos que uno de los árboles es un vértice aislado en la esquina inferior derecha. El subgrafo de la Figura 2(b) tiene tres árboles negros- piezas y un árbol-pieza gris: un vértice aislado en la esquina superior derecha, y dos bordes individuales. Estos cuentan como tres árboles-piezas, a pesar de que vienen del mismo árbol trasero cuando el Gráfico completo considerado. La figura 2 c) muestra otro subgráfico; en este caso hay tres piezas de árboles grises y una negra. En el cuadro 1 figura la terminología de descomposición utilizada en este documento. El problema de descomposición. Definimos el problema de descomposición para gráficos escasos como tak- • un gráfico como su entrada y producción como salida, una descomposición que se puede utilizar para certificar sity. En el presente documento se estudiarán tres tipos de productos: mapas y árboles; descomposiciones adecuadas de `Tk; y la descomposición de guijarros-juego-con-colores, que se define en la siguiente sección. 2. Antecedentes históricos Los conocidos teoremas de Tutte [23] y Nash-Williams [17] relacionan los gráficos (k,k) ajustados a la existencia de descomposicións en los árboles que se extienden por los bordes. Tomando un punto de vista matroidal, 4 Ileana Streinu, Louis Theran Fig. 2. (a) Un gráfico con una descomposición 3T2; uno de los tres árboles es un único vértice en la parte inferior derecha esquina. (b) El subgrafo resaltado dentro del conteo rayado tiene tres piezas de árbol negro y una gris pieza de árbol. (c) El subgrafo resaltado dentro del conteo rayado tiene tres piezas de árbol grises (uno es un solo vértice) y una pieza de árbol negro. Edmonds [3, 4] dio otra prueba de este resultado usando uniones de matroide. La equivalencia de los mapas- los gráficos y árboles y los gráficos ajustados en el rango inferior se muestran utilizando uniones de los matroides en [24], y rutas de aumento matroide son la base de los algoritmos para el rango inferior de [5, 6, 19]. En la teoría de la rigidez un teorema fundacional de Laman [11] muestra que (2,3)-ajustado (Laman) los gráficos corresponden a marcos de barras y conjuntos genéricamente mínimamente rígidos en el plano. Tay [21] ha demostrado ser un resultado análogo para los marcos de la barra del cuerpo en cualquier dimensión utilizando (k,k) gráficos. Rigidez por conteos de interés motivado en el rango superior, y Crapo [2] probó la equivalencia de gráficos Laman y gráficos 3T2 apropiados. Tay [22] utilizó esta condición para dar un prueba directa del teorema de Laman y generalizada la condición 3T2 a todos `Tk para k≤ 2k−1. Haas [7] estudió detalladamente las descomposicións de `Tk y demostró la equivalencia de gráficos ajustados y gráficos `Tk apropiados para el rango superior general. Observamos que aparte de nuestro nuevo guijarro... game-with-colors descomposición, todas las caracterizaciones combinatoria de la gama superior de Los gráficos escasos, incluidos los conteos, tienen una interpretación geométrica [11, 21, 22, 24]. Un algoritmo de juego de guijarros fue propuesto por primera vez en [10] como una alternativa elegante a Hendrick- algoritmos de gráfico Laman de hijo [9]. Berg y Jordania [1], facilitaron el análisis formal de la juego de guijarros de [10] e introdujo la idea de jugar el juego en un gráfico dirigido. Lee y Streinu [12] generalizó el juego de guijarros a toda la gama de parámetros 0≤ 2k−1, y izquierda como un problema abierto utilizando el juego de guijarros para encontrar la escasez certificando las descomposicións. 3. El juego de guijarros con colores Nuestro juego de guijarros con colores es un conjunto de reglas para la construcción de gráficos indexados por no negativos enteros k y `. Usaremos el juego de guijarros con colores como la base de un algoritmo eficiente para el problema de descomposición más adelante en este documento. Puesto que la frase “con colores” es necesaria Sólo en comparación con [12], lo omitiremos en el resto del documento cuando el contexto sea claro. Descomposiciones del gráfico de certificación de la sparsity 5 Ahora presentamos el juego de guijarros con colores. El juego es jugado por un solo jugador en un conjunto finito fijo de vértices. El jugador hace una secuencia finita de movimientos; un movimiento consiste en el adición y/o orientación de un borde. En cualquier momento, el estado del juego es capturado por un gráfico dirigido H, con guijarros de colores sobre vértices y bordes. Los bordes de H son de color por los guijarros en ellos. Mientras que jugando el juego de guijarros todos los bordes están dirigidos, y utilizamos el notación vw para indicar un borde dirigido de v a w. Describimos el juego de guijarros con colores en términos de su configuración inicial y el permitido se mueve. Fig. 3. Ejemplos de juego de guijarros con movimientos de colores: (a) add-edge. b) Deslizamiento de guijarros. Guijarros sobre vértices se muestran como puntos negros o grises. Los bordes están coloreados con el color de la rocalla en ellos. Inicialización: Al principio del juego de guijarros, H tiene n vértices y no tiene bordes. Comenzamos colocando k guijarros en cada vértice de H, uno de cada color ci, para i = 1,2,...,k. Add-edge-with-colors: Dejar v y w ser vértices con al menos â € 1 guijarros en ellos. Asumir (w.l.o.g.) que v tiene al menos un guijarro en él. Recoger un guijarro de v, añadir el borde orientado vw a E(H) y poner el guijarro recogido de v en el nuevo borde. La Figura 3(a) muestra ejemplos del movimiento de add-edge. Pebble-slide: Dejar w ser un vértice con un guijarro p en él, y dejar vw ser un borde en H. Reemplazar vw con wv en E(H); poner el guijarro que estaba en vw en v; y poner p en wv. Tenga en cuenta que el color de un borde puede cambiar con un movimiento de guijarros. La figura 3 b) muestra ejemplos. La convención en estas figuras, y a lo largo de este documento, es que los guijarros sobre los vértices se representan como puntos de color, y que los bordes se muestran en el color de la rocalla en ellos. A partir de la definición del movimiento de guijarros-deslizamiento, es fácil ver que un guijarro en particular es siempre en el vértice donde empezó o en un borde que tiene este vértice como la cola. Sin embargo, al hacer una secuencia de movimientos de guijarros que invierten la orientación de un camino en H, es a veces es conveniente pensar en esta secuencia de inversión del camino como trayendo un guijarro desde el final del camino al principio. La salida de jugar el juego de guijarros es su configuración completa. Salida: Al final del juego, obtenemos el gráfico dirigido H, junto con la ubicación y los colores de los guijarros. Observe que ya que cada borde tiene exactamente un guijarro en él, el guijarro la configuración del juego colorea los bordes. Decimos que el gráfico G de H subyacente no dirigido es construido por el juego (k, `)-pebble o que H es un gráfico de juego de guijarros. Puesto que cada borde de H tiene exactamente un guijarro, las particiones de configuración del juego de guijarro los bordes de H, y así G, en k diferentes colores. Llamamos a esta descomposición de H un guijarro... juego-con-colores descomposición. La Figura 4(a) muestra un ejemplo de un gráfico ajustado (2,2) con un Descomposición de juego de guijarros. Que G = (V,E) sea gráfico de juego de guijarros con la coloración inducida por los guijarros en los bordes, y dejar que G′ sea un subgrafo de G. Entonces la coloración de G induce un conjunto de con- 6 Ileana Streinu, Louis Theran a) b) c) Fig. 4. A (2,2)-término gráfico con una posible descomposición del juego de guijarros. Los bordes están orientados a mostrar (1,0)-esparsidad para cada color. a) El gráfico K4 con una descomposición del juego de guijarros. Hay un árbol negro vacío en el vértice central y un árbol gris que se extiende. b) El subgráfico resaltado consta de dos: árboles negros y un árbol gris; los bordes negros son parte de un ciclo más grande pero aportan un árbol al subgrafo. c) El subgrafo resaltado (con fondo gris claro) tiene tres árboles grises vacíos; los bordes negros contienen un ciclo y no aportan un pedazo de árbol al subgrafo. Significado de la notación longitud (V ′) Número de bordes que se extienden en H por V ′ V ; es decir, EH(V ′) Peb(V ′) Número de guijarros en V ′ ́V fuera (V ′) Número de bordes vw en H con v ́V ′ y w ́V −V ′ pebi(v) Número de guijarros de color ci en v • V outi(v) Número de bordes vw coloreados ci para v â € € TM V Cuadro 2 Pebble notación de juego utilizado en este papel. Subgrafías de G′ (puede haber más de uno del mismo color). Tan monocromático subgraph se llama un mapa-foto-pieza de G′ si contiene un ciclo (en G′) y un árbol-pieza de G′ De lo contrario. El conjunto de piezas arbóreas de G′ es la colección de piezas arbóreas inducidas por G′. Al igual que con la definición correspondiente para `Tk s, el conjunto de piezas arbóreas se define en relación con un sub- grafo; en particular, una pieza de árbol puede formar parte de un ciclo más grande que incluye bordes que no se extienden por G′. Las propiedades de las descomposicións del juego de guijarros se estudian en la Sección 6 y en el Teorema 2 muestra que cada color debe ser (1,0)-sparse. La orientación de los bordes en la Figura 4(a) muestra Esto. Por ejemplo, la Figura 4(a) muestra un gráfico ajustado (2,2) con un posible decom de juego de guijarro- posición. El gráfico completo contiene una pieza de árbol gris y una pieza de árbol negro que es un aislado vértice. El subgrafo de la Figura 4(b) tiene un árbol negro y un árbol gris, con los bordes del negro árbol procedente de un ciclo en el gráfico más grande. En la Figura 4(c), sin embargo, el ciclo negro no contribuir con una pieza de árbol. Las tres piezas de árbol en este subgrafo son árboles grises de un solo vértex. En la siguiente discusión, utilizamos la notación peb(v) para el número de guijarros en v y pebi(v) para indicar el número de guijarros de colores i en v. La Tabla 2 enumera la notación de juego de guijarros utilizada en este artículo. 4. Nuestros resultados Describimos nuestros resultados en esta sección. El resto del periódico proporciona las pruebas. Descomposiciones del gráfico de certificación de la sparsity 7 Nuestro primer resultado es un fortalecimiento de los juegos de guijarros de [12] para incluir los colores. Dice que los gráficos escasos son exactamente gráficos de juego de guijarros. Recuerde que a partir de ahora, todos los juegos de guijarros discutidos en este artículo son nuestro juego de guijarros con colores a menos que se anote explícitamente. Teorema 1 (Los gráficos Sparse y los gráficos de juego de guijarros coinciden). Un gráfico G es (k, `)-sparse con 0≤ 2k−1 si y sólo si G es un gráfico de juego de guijarros. A continuación consideramos las descomposiciones de juego de guijarros, mostrando que son una generalización de las descomposiciones adecuadas de `Tk que se extienden a toda la gama matroidal de gráficos dispersos. Teorema 2 (La descomposición de guijarros-juego-con-colores). Un gráfico G es un juego de guijarros gráfico si y sólo si admite una descomposición en k borde-discoint subgraphs tales que cada uno es (1,0)-sparse y cada subgrafo de G contiene al menos ` piezas de árbol de la (1,0)-sparse gráficos en la descomposición. Las subgrafías de (1,0)-parse en la declaración de Teorema 2 son los colores de los guijarros; por lo tanto Teorema 2 da una caracterización de las descomposicións de guijarros-juego-con-colores obtenidos jugando el juego de guijarros definido en la sección anterior. Nótese la similitud entre el requisito de que el conjunto de piezas arbóreas tenga por lo menos un tamaño ` en el Teorema 2 y la definición de un propiamente dicho `Tk. Nuestros siguientes resultados muestran que para cualquier gráfico de juego de guijarros, podemos especializar su juego de guijarros construcción para generar una descomposición que es un mapa-y-árboles o `Tk. Nosotros llamamos a estos especializada construcción de juegos de guijarros canónicos, y el uso canónico juego de guijarros construc- ciones, obtenemos nuevas pruebas directas de los resultados de arboricidad existentes. Observamos Teorema 2 que los mapas-y-árboles son casos especiales del juego de guijarros decompo- Situación: tanto los árboles que se extienden y los mapas que se extienden son (1.0)-parse, y cada uno de la extensión los árboles aportan al menos un pedazo de árbol a cada subgrafo. El caso de los gráficos `Tk apropiados es más sutil; si cada color en una descomposición del juego de guijarros es un bosque, entonces hemos encontrado un adecuado `Tk, pero esta clase es un subconjunto de todos los posibles apropiados `Tk descomposiciones de un gráfico apretado. Demostramos que esta clase de descomposiciones apropiadas `Tk es suficiente para certificar la escasez. Ahora declaramos el teorema principal para el rango superior e inferior. Teorema 3 (Teorema Principal): Mapas y árboles coinciden con el juego de guijarros grafos). Que 0 ≤ ` ≤ k. Un gráfico G es un gráfico de juego de guijarro apretado si y sólo si G es un (k, `)- mapas y árboles. Teorema 4 (Teorema principal): Los gráficos `Tk adecuados coinciden con el juego de guijarros grafos). Deje k≤ 2k−1. Un gráfico G es un gráfico de juego de guijarros apretado si y sólo si es un adecuado `Tk con kn− ` bordes. Como corolarios, obtenemos los resultados de descomposición existentes para gráficos escasos. Corollario 5 (Nash-Williams [17], Tutte [23], White y Whiteley [24]). Deja k. Un gráfico G es estrecho si y sólo si tiene una descomposición (k, `)-maps-and-trees. Corollario 6 (Crapo [2], Haas [7]). Dejar k ≤ 2k−1. Un gráfico G es estrecho si y sólo si es un propiamente dicho `Tk. Encontrar eficientemente construcciones canónicas de juego de guijarros. Las pruebas de Teorema 3 y Theo- rem 4 conduce a un algoritmo obvio con O(n3) tiempo de ejecución para el problema de descomposición. Nuestro último resultado mejora en esto, mostrando que una construcción canónica juego de guijarros, y por lo tanto 8 Ileana Streinu, Louis Theran un mapa-y-árboles o `Tk descomposición apropiada se puede encontrar usando un algoritmo de juego de guijarros en O(n2) tiempo y espacio. Estos límites de tiempo y espacio significan que nuestro algoritmo puede combinarse con los de [12] sin ningún cambio en la complejidad. 5. Gráficos de juego de pebble En esta sección demostramos Teorema 1, un fortalecimiento de los resultados de [12] al juego de guijarros con colores. Dado que muchas de las propiedades relevantes del juego de guijarros con colores directamente de los juegos de guijarros de [12], nos referimos al lector allí para las pruebas. Comenzamos estableciendo algunas invariantes que se mantienen durante la ejecución del juego de guijarros. Lemma 7 (invariantes de juego de pebble). Durante la ejecución del juego de guijarros, lo siguiente los invariantes se mantienen en H: (I1) Hay por lo menos ` guijarros en V. [12] (I2) Para cada vértice v, span(v)+out(v)+peb(v) = k. [12] (I3) Para cada V ′ ́V, span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) Por cada vértice v V, outi(v)+pebi(v) = 1. (I5) Cada ruta máxima que consiste sólo de bordes con ci de color termina en el primer vértice con un guijarro de color ci o un ciclo. Prueba. (I1), (I2), y (I3) vienen directamente de [12]. (I4) Este invariante se mantiene claramente en la fase de inicialización del juego de guijarros con colores. Esa reserva de movimientos de bordes añadidos y guijarros (I4) está clara de la inspección. (I5) Por (I4), un camino monocromático de los bordes se ve obligado a terminar sólo en un vértice con un guijarro de el mismo color en ella. Si no hay guijarros de ese color alcanzable, entonces el camino debe eventualmente Visita un vértice dos veces. De estos invariantes, podemos mostrar que los gráficos constructibles del juego de guijarros son escasos. Lemma 8 (Los gráficos de los juegos de pelota son escasos [12]). Dejar H ser un gráfico construido con el Juego de guijarros. Entonces H es escasa. Si hay exactamente ` guijarros en V (H), entonces H es apretado. El paso principal para probar que cada gráfico escaso es un gráfico de juego de guijarros es el siguiente. Recordemos que al traer un guijarro a v nos referimos a reorientar H con movimientos de guijarro-deslizamiento para reducir el grado de v por uno. Lemma 9 (La condición de guijarro â € 1 [12]). Dejar vw ser un borde tal que H + vw es escaso. Si peb({v,w}) < â € 1, entonces un guijarro no en {v,w} se puede llevar a v o w. Se deduce que cualquier gráfico escaso tiene una construcción de juego de guijarros. Teorema 1 (Los gráficos Sparse y los gráficos de juego de guijarros coinciden). Un gráfico G es (k, `)-sparse con 0≤ 2k−1 si y sólo si G es un gráfico de juego de guijarros. 6. La descomposición de guijarros-juego-con-colores En esta sección demostramos Teorema 2, que caracteriza todas las descomposicións de juego de guijarros. Nosotros empezar con los siguientes lemas sobre la estructura de los componentes monocromáticos conectados en H, el gráfico dirigido mantenido durante el juego de guijarros. Descomposiciones del gráfico de certificación de la sparsity 9 Lemma 10 (los subgrafos monocromáticos del juego de guijarros son (1,0)-sparse). Deja que Hi sea el sub- gráfico de H inducido por los bordes con guijarros de color ci en ellos. Entonces Hi es (1,0)-parso, para i = 1,...,k. Prueba. Por (I4) Hi es un conjunto de bordes con grado a lo sumo uno para cada vértice. Lemma 11 (Piezas de árbol en un gráfico de juego de guijarros). Cada subgrafo del gráfico dirigido H en una construcción de juego de guijarros contiene por lo menos ` piezas monocromáticas de árboles, y cada uno de estos tiene sus raíces en un vértice con un guijarro en él o un vértice que es la cola de un borde. Recordemos que un borde superior a un subpárrafo H ′ = (V ′,E ′) es un borde vw con v′ V y vw /′ E. Prueba. Dejar que H ′ = (V ′,E ′) sea un subgrafo no vacío de H, y asumir sin pérdida de generalidad que H ′ es inducida por V ′. Por (I3), fuera (V ′)+ peb(V ′) ≥ `. Mostraremos que cada guijarro y cola de borde es la raíz de una pieza de árbol. Considerar un vértice v V ′ y un color ci. Por (I4) hay un único monocromático dirigido ruta de color ci a partir de v. Por (I5), si este camino termina en una rocalla, no tiene un ciclo. Del mismo modo, si este camino alcanza un vértice que es la cola de un borde también en color ci (es decir, si el trayectoria monocromática desde v hojas V ′), entonces la trayectoria no puede tener un ciclo en H ′. Dado que este argumento funciona para cualquier vértice en cualquier color, para cada color hay una partición de los vértices en aquellos que pueden alcanzar cada guijarro, cola de borde superior, o ciclo. De ello se deduce que cada uno de guijarros y cola de borde superior es la raíz de un árbol monocromático, como se desee. Aplicado a todo el gráfico Lemma 11 nos da lo siguiente. Lemma 12 (Los pebbles son las raíces de los árboles). En cualquier configuración de juego de guijarros, cada guijarros de color ci es la raíz de un (posiblemente vacío) monocromático árbol-pieza de color ci. Nota: Haas mostró en [7] que en un `Tk, un subgráfico inducido por n′ ≥ 2 vértices con m′ los bordes tienen exactamente piezas de árbol knm′ en él. Lemma 11 refuerza el resultado de Haas al ampliarlo a la gama inferior y dando una construcción que encuentra las piezas de árbol, mostrando la conexión entre la condición de guijarro â € 1 y la condición hereditaria en la adecuada `Tk. Concluimos nuestra investigación de construcciones arbitrarias de juego de guijarros con una descripción de la descomposición inducida por el juego de guijarros con colores. Teorema 2 (La descomposición de guijarros-juego-con-colores). Un gráfico G es un juego de guijarros gráfico si y sólo si admite una descomposición en k borde-discoint subgraphs tales que cada uno es (1,0)-sparse y cada subgrafo de G contiene al menos ` piezas de árbol de la (1,0)-sparse gráficos en la descomposición. Prueba. Deja que G sea un gráfico de juego de guijarros. La existencia de la k borde-disjunta (1,0)-sparse sub- Los gráficos fueron mostrados en Lemma 10, y Lemma 11 prueba la condición en subgrafías. Para la otra dirección, observamos que un ci de color con piezas de árbol ti en un subgrafo dado puede espacio a lo sumo n- ti bordes; sumando sobre todos los colores muestra que un gráfico con un guijarro-juego la descomposición debe ser escasa. Aplique el Teorema 1 para completar la prueba. Observación: Observamos que una descomposición del juego de guijarros para un gráfico de Laman puede ser leída de la coincidencia bipartita utilizada en el algoritmo de extracción de gráficos Laman de Hendrickson [9]. De hecho, las orientaciones de juego de guijarros tienen una correspondencia natural con los emparejamientos bipartitos utilizados en 10 Ileana Streinu, Louis Theran Mapas y árboles son un caso especial de descomposición de juegos de guijarros para gráficos apretados: si hay no son ciclos en ` de los colores, entonces los árboles enraizados en los ` guijarros correspondientes deben ser que se extienden, ya que tienen n - 1 bordes. Además, si cada color forma un bosque en un rango superior la descomposición del juego de guijarros, entonces la condición de piezas de árbol asegura que el juego de guijarros de- la composición es un `Tk. En la siguiente sección, mostramos que el juego de guijarros puede ser especializado para corresponder a los mapas- y árboles y las correspondientes descomposicións `Tk. 7. Construcciones Canónicas de Juego de Pebble En esta sección demostramos los principales teoremas (Teorema 3 y Teorema 4), continuando las inves- de las descomposiciones inducidas por las construcciones de juego de guijarros mediante el estudio del caso en el que un Se crea un número mínimo de ciclos monocromáticos. La idea principal, capturada en Lemma 15 e ilustrado en la Figura 6, es evitar la creación de ciclos al recoger piedras. Demostramos que esto es siempre posible, lo que implica que los mapas monocromáticos se crean sólo cuando añadir más de k(n1) bordes a algún conjunto de n′ vértices. Para el rango inferior, esto implica que Cada color es un bosque. Cada caracterización de descomposición de gráficos ajustados discutidos arriba sigue inmediatamente del teorema principal, dando nuevas pruebas de los resultados anteriores en un un marco unificado. En la prueba, vamos a utilizar dos especializaciones de los movimientos de juego de guijarros. El primero es un modi- ficación del movimiento de add-edge. Add-edge canónico: Al realizar un movimiento de add-edge, cubra el nuevo borde con un color que está en ambos vértices si es posible. Si no, entonces tome el color numerado más alto presente. La segunda es una restricción en la que los movimientos de guijarros-deslizamiento que permitimos. Deslizamiento canónico de guijarros: Un movimiento de guijarros se permite sólo cuando no crea un ciclo monocromático. Llamamos a una construcción de juego de guijarros que utiliza sólo estos movimientos canónicos. En esta sección vamos a mostrar que cada gráfico de juego de guijarros tiene una construcción canónica de juego de guijarros (Lemma 14 y Lemma 15) y que las construcciones canónicas de juego de guijarros corresponden a `Tk y las descomposicións de mapas y árboles (Teorema 3 y Teorema 4). Comenzamos con un lema técnico que motiva la definición de juego canónico de guijarros construcciones. Muestra que las situaciones desaprobadas por los movimientos canónicos son todas las maneras para que los ciclos se formen en los colores más bajos. Lemma 13 (creación del ciclo monocromático). Let v â € ¢ V tener un guijarro p de color ci en él y dejar w ser un vértice en el mismo árbol de color ci como v. Un ciclo monocromático de color ci se crea exactamente de una de las siguientes maneras: (M1) El borde vw se añade con un movimiento de add-edge. (M2) El borde wv es invertido por un movimiento de guijarro-deslizamiento y el guijarro p se utiliza para cubrir el reverso edge vw. Prueba. Observe que las condiciones previas en la declaración del lema están implícitas en Lemma 7. Por Lemma 12 ciclos monocromáticos se forman cuando el último guijarro de color ci se elimina de un Subgrafía monocromática conectada. (M1) y (M2) son las únicas maneras de hacer esto en un guijarro construcción del juego, ya que el color de un borde sólo cambia cuando se inserta la primera vez o un guijarro nuevo es puesto en él por un movimiento de guijarro-deslizamiento. Descomposiciones del gráfico de certificación de la sparsity 11 vw vw Fig. 5. Crear ciclos monocromáticos en un juego (2.0)-pebble. a) Un movimiento de tipo (M1) crea un ciclo por añadir un borde negro. (b) Un movimiento de tipo (M2) crea un ciclo con un movimiento de guijarros-deslizamiento. Los vértices son etiquetado de acuerdo a su papel en la definición de los movimientos. La figura 5 a) y la figura 5 b) muestran ejemplos de movimientos de creación de mapas (M1) y (M2), respectivamente, en una construcción de juego (2.0)-pebble. A continuación mostramos que si un gráfico tiene una construcción de juego de guijarros, entonces tiene un peb canónico- ble construcción de juegos. Esto se hace en dos pasos, considerando los casos (M1) y (M2) sepa- - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. La prueba da dos construcciones que implementan el add-edge canónico y canónico movimiento de guijarros-deslizamiento. Lemma 14 (El movimiento canónico de add-edge). Let G ser un gráfico con un juego de guijarros construc- tion. Los pasos de creación de ciclo de tipo (M1) se pueden eliminar en colores ci para 1 ≤ i ≤, donde = min{k,. Prueba. Para los movimientos de add-edge, cubra el borde con un color presente en v y w si es posible. Si esto no es posible, entonces hay â € 1 colores distintos presentes. Usar el color numerado más alto para cubrir el nuevo borde. Observación: Observamos que en el rango superior, siempre hay un color repetido, por lo que no canónico los movimientos de add-edge crean ciclos en el rango superior. El movimiento canónico de guijarros se define por una condición global. Para demostrar que obtenemos la misma clase de gráficos usando sólo movimientos canónicos de rocalla-deslizamiento, tenemos que extender Lemma 9 a sólo movimientos canónicos. El paso principal es mostrar que si hay alguna secuencia de movimientos que reorienta un camino de v a w, entonces hay una secuencia de movimientos canónicos que hace lo mismo Cosa. Lemma 15 (El movimiento canónico de guijarros). Cualquier secuencia de deslizamiento de guijarros se mueve llevando a un movimiento de add-edge se puede reemplazar por uno que no tiene pasos (M2) y permite el mismo add-edge move. En otras palabras, si es posible recoger 1 guijarros en los extremos de un borde a añadir, entonces es posible hacer esto sin crear ningún ciclo monocromático. 12 Ileana Streinu, Louis Theran La Figura 7 y la Figura 8 ilustran la construcción utilizada en la prueba de Lemma 15. Nosotros llamamos a esto la construcción de atajos por analogía a la unión matroide y caminos de aumento de intersección utilizados en trabajos anteriores en el rango inferior. La Figura 6 muestra la estructura de la prueba. La construcción de acceso directo elimina un paso (M2) al principio de una secuencia que reorienta un camino de v a w con deslizamientos de guijarros. Desde uno la aplicación de la construcción abreviada reorienta un camino simple de un vértice w′ a w, y un ruta de v a w′ se conserva, la construcción de acceso directo se puede aplicar inductivamente para encontrar la secuencia de movimientos que queremos. Fig. 6. Esquema de la construcción del atajo: (a) Un camino sencillo arbitrario de v a w con líneas curvas indicando caminos simples. b) Una etapa (M2). El borde negro, a punto de ser volteado, crearía un ciclo, se muestra en gris rayado y sólido, del (único) árbol gris enraizado en w. Los bordes grises sólidos eran parte de la ruta original de (a). (c) El camino acortado a la rocalla gris; el nuevo camino sigue el gris árbol todo el camino desde la primera vez que el camino original tocó el árbol gris en w′. La ruta de v a w′ es simple, y la construcción del atajo se puede aplicar inductivamente a él. Prueba. Sin pérdida de generalidad, podemos asumir que nuestra secuencia de movimientos reorienta un simple camino en H, y que el primer movimiento (el final del camino) es (M2). El paso (M2) mueve un guijarro de color ci de un vértice w en el borde vw, que se invierte. Porque el movimiento es (M2), v y w están contenidos en un árbol monocromático máximo de color ci. Llame a este árbol H ′i, y observar que está arraigado en w. Ahora considere los bordes invertidos en nuestra secuencia de movimientos. Como se ha señalado anteriormente, antes de hacer cualquiera de los movimientos, estos bosquejan un camino simple en H que termina en w. Que z sea el primer vértice en este camino en H ′i. Modificamos nuestra secuencia de movimientos de la siguiente manera: eliminar, desde el principio, cada mover antes de la que invierte algunos yz borde; prepend en lo que queda una secuencia de movimientos que mueve el guijarro en w a z en H ′i. Descomposiciones del gráfico de certificación de la sparsity 13 Fig. 7. Eliminando movimientos (M2): (a) un movimiento (M2); (b) evitando el (M2) moviéndose por otro camino. El camino donde se mueven los guijarros está indicado por líneas duplicadas. Fig. 8. Eliminación (M2) movimientos: (a) el primer paso para mover el guijarro negro a lo largo del camino doble es (M2); (b) evitando el (M2) y simplificando el camino. Puesto que ningún borde cambia de color en el comienzo de la nueva secuencia, hemos eliminado el movimiento (M2). Porque nuestra construcción no cambia ninguno de los bordes involucrados en el cola restante de la secuencia original, la parte de la ruta original que queda en el nuevo secuencia seguirá siendo un camino simple en H, cumpliendo con nuestra hipótesis inicial. El resto del lema sigue por inducción. Juntos Lemma 14 y Lemma 15 prueban lo siguiente. Lemma 16. Si G es un gráfico de juego de guijarros, entonces G tiene una construcción canónica de juego de guijarros. Usando construcciones canónicas de juego de guijarros, podemos identificar los gráficos apretados de juego de guijarros con mapas y árboles y gráficos `Tk. 14 Ileana Streinu, Louis Theran Teorema 3 (Teorema Principal): Mapas y árboles coinciden con el juego de guijarros grafos). Que 0 ≤ ` ≤ k. Un gráfico G es un gráfico de juego de guijarro apretado si y sólo si G es un (k, `)- mapas y árboles. Prueba. Como se observó anteriormente, una descomposición de mapas y árboles es un caso especial del juego de guijarros descomposición. Aplicando el Teorema 2, vemos que cualquier mapa y árbol debe ser un juego de guijarros gráfico. Para la dirección inversa, considere la construcción canónica de un juego de guijarros de un gráfico apretado. Desde Lemma 8, vemos que quedan piedras en G al final de la construcción. Los definición del movimiento canónico de add-edge implica que debe haber al menos un guijarro de cada ci para i = 1,2,........................................................................................................... Se deduce que hay exactamente uno de cada uno de estos colores. Por Lemma 12, cada uno de estos guijarros es la raíz de una pieza arbórea monocromática con n - 1 bordes, dando los árboles de separación de bordes necesarios. Corollario 5 (Nash-Williams [17], Tutte [23], White y Whiteley [24]). Deja k. Un gráfico G es estrecho si y sólo si tiene una descomposición (k, `)-maps-and-trees. A continuación consideramos las descomposicións inducidas por las construcciones canónicas de juego de guijarros cuando k +1. Teorema 4 (Teorema Principal): Árboles y árboles adecuados coinciden con el ble-game graphs). Deje k≤ 2k−1. Un gráfico G es un gráfico de juego de guijarro apretado si y sólo si es un `Tk con bordes kn− ` adecuado. Prueba. Como se ha señalado anteriormente, una descomposición adecuada de `Tk debe ser escasa. Lo que tenemos que mostrar es que una construcción canónica de un juego de guijarros de un gráfico apretado produce una adecuada `Tk. Por Teorema 2 y Lemma 16, ya tenemos la condición en los árboles-piezas y el decom- posición en `árboles de borde-desconectado. Por último, una aplicación de (I4), muestra que cada vértice debe en exactamente k de los árboles, según sea necesario. Corollario 6 (Crapo [2], Haas [7]). Dejar k ≤ 2k−1. Un gráfico G es estrecho si y sólo si es un propiamente dicho `Tk. 8. Algoritmos de juego de pebble para encontrar descomposicións Una ejecución naïve de las construcciones en la sección anterior conduce a un algoritmo re- tiempo para recoger cada guijarro en una construcción canónica: en el peor de los casos aplicaciones de la construcción en Lemma 15 requiriendo tiempo cada uno, dando un total de ejecución tiempo de فارسى(n3) para el problema de descomposición. En esta sección, describimos algoritmos para el problema de descomposición que se ejecutan en el tiempo O(n2). Comenzamos con la estructura general del algoritmo. Algoritmo 17 (El juego canónico de guijarros con colores). Entrada: Un gráfico G. Salida: Un gráfico de juego de guijarros H. Método: – Conjunto V (H) = V (G) y colocar un guijarro de cada color en los vértices de H. – Para cada borde vw E(G) tratar de recoger al menos 1 guijarros en v y w utilizando guijarros deslizante movimientos según lo descrito por Lemma 15. Descomposiciones del gráfico de certificación de la Sparsity 15 – Si al menos 1 guijarros se puede recoger, añadir vw a H utilizando un movimiento de borde añadido como en Lemma 14, por lo demás descarte vw. – Finalmente, devolver H, y las ubicaciones de los guijarros. Correcto. Teorema 1 y el resultado de [24] que los gráficos escasos son los independientes conjuntos de un matroide muestran que H es un subgrafo de tamaño máximo escaso de G. Desde la construcción encontrado es canónico, el teorema principal muestra que el color de los bordes en H da un mapa- y-árboles o descomposición adecuada `Tk. Complejidad. Comenzamos observando que el tiempo de ejecución del Algoritmo 17 es el tiempo necesario para proceso O(n) bordes añadidos a H y O(m) bordes no añadidos a H. Primero consideramos el costo de un borde de G que se añade a H. Cada uno de los movimientos de juego de guijarros se puede implementar en tiempo constante. Lo que queda es a describir una manera eficiente de encontrar y mover los guijarros. Utilizamos el siguiente algoritmo como un Subrutina de Algoritmo 17 para hacer esto. Algoritmo 18 (Encontrar un camino canónico a una rocalla.). Entrada: Vertices v y w, y una configuración de juego de guijarros en un gráfico dirigido H. Salida: Si se encontró un guijarro, ‘sí’ y ‘no’ de otra manera. Se actualiza la configuración de H. Método: – Comience por hacer una búsqueda de profundidad desde v en H. Si no se encuentra ningún guijarro en w, detener y devolver «no.» – De lo contrario se encontró un guijarro. Ahora tenemos una ruta v = v1,e1,. ..,ep−1,vp = u, donde el vi son vértices y ei es el borde vivi+1. Que c[ei] sea el color del guijarro en ei. Usaremos la matriz c[] para hacer un seguimiento de los colores de los guijarros en los vértices y los bordes después de moverlos y el array s[] para dibujar un camino canónico de v a u encontrando un sucesor para cada uno borde. – Establecer s[u] = «end′ y establecer c[u] al color de una piedra arbitraria en u. Caminamos en el camino en orden inverso: vp,ep−1,ep−2,. ..,e1,v1. Para cada i, verifique si c[vi] está configurado; si es así, vaya a la siguiente i. De lo contrario, compruebe si c[vi+1] = c[ei]. – Si lo es, establece s[vi] = ei y establece c[vi] = c[ei], y pasa al siguiente borde. – De lo contrario c[vi+1] 6= c[ei], tratar de encontrar un camino monocromático en color c[vi+1] de vi a vi+1. Si un vértice x se encuentra para el cual c[x] se establece, tenemos una ruta vi = x1, f1,x2,. .., fq−1,xq = x que es monocromático en el color de los bordes; establecer c[xi] = c[fi] y s[xi] = fi para i = 1,2,...,q−1. Si c[x] = c[ fq−1], pare. De lo contrario, comprobar recursivamente que no hay un monocro- c[x] ruta mática de xq−1 a x usando este mismo procedimiento. – Finalmente, deslizar guijarros a lo largo del camino desde los puntos finales originales v a u especificado por el array sucesor s[v], s[s[v],... La corrección de Algoritmo 18 viene del hecho de que está implementando el atajo construcción. La eficiencia viene del hecho de que en lugar de potencialmente mover el guijarro hacia atrás y adelante, Algoritmo 18 pre-computa un camino canónico que cruza cada borde de H a lo sumo tres times: una vez en la primera búsqueda de profundidad inicial, y dos veces al convertir la ruta inicial a una Canónico. De ello se deduce que cada borde aceptado toma O(n) tiempo, para un total de O(n2) tiempo los bordes de procesamiento gastados en H. Aunque no hemos discutido esta explicitación, para que el algoritmo sea eficiente necesitamos mantener los componentes como en [12]. Después de cada borde aceptado, los componentes de H se pueden actualizar en el tiempo O(n). Por último, los resultados de [12, 13] muestran que los bordes rechazados toman un O(1) amortizado tiempo cada uno. 16 Ileana Streinu, Louis Theran Resumiendo, hemos demostrado que el juego canónico de guijarros con colores resuelve la decom- problema de posición en el tiempo O(n2). 9. Un caso especial importante: Rigidez en la dimensión 2 y slider-pinning En esta breve sección presentamos una nueva solicitud para el caso especial de importancia práctica, k = 2, ` = 3. Como se explica en la introducción, el teorema de Laman [11] caracteriza mínimamente gráficos rígidos como los gráficos ajustados (2,3). En el trabajo reciente sobre el slider pinning, desarrollado después de la El documento actual fue presentado, introdujimos el modelo de slider-pinning de rigidez [15, 20]. Com- binatoriamente, modelamos los marcos bar-slider como gráficos simples junto con algunos bucles colocados en sus vértices de tal manera que no haya más de 2 bucles por vértice, uno de cada uno color. Caracterizamos los gráficos de deslizadores de barras mínimamente rígidos [20] como gráficos que son: 1. (2,3)-parse para subgrafías que no contengan bucles. 2. (2,0)-ajustado cuando se incluyen los bucles. Llamamos a estos gráficos (2,0,3)-clasificados-ajustados, y son un caso especial de la clasificación-parse gráficos estudiados en nuestro artículo [14]. La conexión con los juegos de guijarros en este artículo es la siguiente. Corollary 19 (juegos de pebble y slider-pinning). En cualquier gráfico de juego (2,3)-pebble, si Reemplazar los guijarros por los bucles, obtenemos un gráfico ajustado (2.0,3)-calificado. Prueba. Seguidos de invariantes (I3) de Lemma 7. En [15], estudiamos un caso especial de slider pinning donde cada slider es vertical o horizontal. Modelamos los deslizadores como bucles precoloreados, con el color que indica la dirección x o y. Para este caso de deslizador paralelo eje, los gráficos mínimamente rígidos se caracterizan por: 1. (2,3)-parse para subgrafías que no contengan bucles. 2. Admitir un 2-coloración de los bordes para que cada color sea un bosque (es decir, no tiene ciclos), y cada uno árbol monocromático abarca exactamente un bucle de su color. Esto también tiene una interpretación en términos de juegos de guijarros de colores. Corollary 20 (El juego de guijarros con colores y slider-pinning). En cualquier canónico (2,3)- Guijarro-juego-con-colores gráfico, si reemplazamos los guijarros por bucles del mismo color, obtenemos el gráfico de un marco de eje-paralelo de barra-slider mínimamente fijado. Prueba. Sigue desde el Teorema 4, y Lemma 12. 10. Conclusiones y problemas pendientes Presentamos una nueva caracterización de (k, `)-sparse gráficos, el juego de guijarros con colores, y lo utilizó para dar un algoritmo eficiente para encontrar descomposicións de gráficos escasos en el borde- árboles desarticulados. Nuestro algoritmo encuentra tales descomposiciones certificadoras de esparcimiento en el rango superior y se ejecuta en el tiempo O(n2), que es tan rápido como los algoritmos para reconocer gráficos escasos en el rango superior a partir de [12]. También usamos el juego de guijarros con colores para describir una nueva descomposición de la sparsity-certificating- ciones que se aplican a toda la gama matroidal de gráficos dispersos. Descomposiciones del gráfico de certificación de la sparsity 17 Definimos y estudiamos una clase de construcciones canónicas de juego de guijarros que corresponden a o bien una descomposición de mapas y árboles o bien una descomposición adecuada de `Tk. Esto da una nueva prueba de la Tutte-Nash- Teorema de arboricidad Williams y una prueba unificada de la descomposición previamente estudiada cer- tificates de la esparzidad. Las construcciones canónicas de juego de guijarros también muestran la relación entre la condición de guijarro â 1, que se aplica a la gama superior de â, para aumentar la unión de los matroides rutas, que no se aplican en el rango superior. Consecuencias algorítmicas y problemas abiertos. En [6], Gabow y Westermann dan un O(n3/2) algoritmo para reconocer gráficos escasos en el rango inferior y extraer subtítulos escasos de Densos. Su técnica se basa en la búsqueda eficiente de caminos de aumento de unión de matroides, que extienden una descomposición de mapas y árboles. El algoritmo O(n3/2) utiliza dos subrutinas para encontrar rutas de aumento: exploración cíclica, que encuentra rutas de aumento uno a la vez, y lote escaneado, que encuentra grupos de caminos de aumento disjuntos. Observamos que Algoritmo 17 se puede utilizar para reemplazar el escaneo cíclico en Gabow y Wester- algoritmo de mann sin cambiar el tiempo de ejecución. Las estructuras de datos utilizadas en la aplicación de guijarros, detallado en [12, 13] son más simples y más fáciles de implementar que los utilizado para apoyar el escaneo cíclico. Los dos principales problemas algorítmicos abiertos relacionados con el juego de guijarros son entonces: Problema 1. Desarrollar un algoritmo de juego de guijarros con las propiedades de escaneado por lotes y obtener un algoritmo O(n3/2) implementable para el rango inferior. Problema 2. Extender la exploración por lotes a la condición de guijarro â € 1 y derivar un guijarro O(n3/2) algoritmo de juego para el rango superior. En particular, sería de importancia práctica encontrar un algoritmo O(n3/2) implementable para las descomposiciones en los árboles que se extienden por los bordes. Bibliografía 1. Berg, A.R., Jordán, T.: Algoritmos para la rigidez gráfica y el análisis de la escena. In: Proc. 11a Simposio Europeo sobre Algoritmos (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: Sobre la rigidez genérica de los marcos planos. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Partición mínima de un matroide en conjuntos independientes. J. Res. Nat. Bur. Normas Secc. B 69B, 67–72 (1965) 4. Edmonds, J.: Funciones submodulares, matroides y ciertos poliedros. En: Combinatoria Optimización: ¡Eureka, encogerte!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: Un enfoque matroide para encontrar conectividad de borde y arborescencias de embalaje. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Bosques, marcos y juegos: Algoritmos para sumas de matroide y aplicaciones. Algoritmica 7(1), 465–497 (1992) 7. Haas, R.: Caracterizaciones de la arboricidad de los gráficos. Ars Combinatoria 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Caracterizando gráficos escasos por mapa decompo- Situaciones. Revista de Matemáticas Combinatoria y Computación Combinatoria 62, 3-11 (2007) 9. Hendrickson, B.: Condiciones para realizaciones gráficas únicas. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: Un algoritmo para la percolación de rigidez bidimensional: la Juego de guijarros. Revista de Física Computacional 137, 346-365 (1997) 11. Laman, G.: En gráficos y rigidez de las estructuras esqueléticas planas. Revista de Ingeniería Matemáticas 4, 331-340 (1970) 12. Lee, A., Streinu, I.: Algorihms de juego de pebble y gráficos escasos. Matemáticas discretas 308(8), 1425-1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Encontrar y mantener componentes rígidos. In: Proc. Cana... Conferencia de Geometría Computacional. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Gráficos bajos y matroides. Diario de Universal Ciencias de la computación 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: El problema del slider-pinning. En: Actas del 19 Conferencia Canadiense sobre Geometría Computacional (CCCG’07) (2007) 16. Lovász, L.: Problemas y ejercicios combinatorios. Akademiai Kiado y North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Descomposición de gráficos finitos en los bosques. Diario de Londres Sociedad Matemática 39, 12 (1964) 18. Oxley, J.G.: Teoría de los matroides. The Clarendon Press, Oxford University Press, Nueva York (1992) 19. Roskind, J., Tarjan, R.E.: Una nota sobre la búsqueda de un coste mínimo borde de árboles disjuntos que se extienden. Matemáticas de la investigación de operaciones 10(4), 701-708 (1985) 20. Streinu, I., Theran, L.: Genericidad combinatoria y rigidez mínima. En: SCG ’08: Pro- cedidas del 24o Simposio anual sobre Geometría Computacional, pp. 365– 374. ACM, Nueva York, NY, USA (2008). 21. Tay, T.S.: Rigidez de los multógrafos I: uniendo cuerpos rígidos en n-espacio. Diario de Combinato- rial Theory, Serie B 26, 95–112 (1984) 22. Tay, T.S.Una nueva prueba del teorema de Laman. Gráficos y combinatorios 9, 365–370 (1993) 23. Tutte, W.T.: Sobre el problema de la descomposición de un gráfico en n factores conectados. Diario de Sociedad Matemática de Londres 142, 221–230 (1961) 24. Whiteley, W.: La unión de los matroides y la rigidez de los marcos. SIAM Journal on Matemáticas discretas 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introducción y preliminares Antecedentes históricos El juego de guijarros con colores Nuestros resultados Gráficos de juego de pebble La descomposición de guijarros-juego-con-colores Construcciones Canónicas de Juego de Pebble Algoritmos de juego de pebble para encontrar descomposicións Un caso especial importante: Rigidez en la dimensión 2 y slider-pinning Conclusiones y problemas pendientes

The evolution of the Earth-Moon system based on the dark matter field fluid model

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"The evolution of the Earth-Moon system based on the dark fluid model\nThe evolution of the Earth-Mo(...TRUNCATED)

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Numerical solution of shock and ramp compression for general material properties

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$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight

" Serre obtained the p-adic limit of the integral Fourier coefficient of\nmodular forms on $SL_2(\\(...TRUNCATED)

"p-ADIC LIMIT OF THE FOURIER COEFFICIENTS OF WEAKLY\nHOLOMORPHIC MODULAR FORMS OF HALF INTEGRAL WEIG(...TRUNCATED)

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