efficient decision procedures for graph properties on context free graph languages efficient ways of analyzing families of graphs that are generated by a certain type of context free graph grammars are considered these graph grammars are called cellular graph grammars they generate the same graph families as hyperedge replacement systems but are defined in a way that supports complexity analysis a characteristic called finiteness of graph properties are defined and combinatorial algorithms are presented for deciding whether a graph language generated by a given cellular graph grammar contains a graph with a given finite graph property structural parameters are introduced that bound the complexity of the decision procedure and special cases for which the decision can be made in polynomial time are discussed extensions to graph grammars that are not context free are also given our results provide explicit and efficient combinatorial algorithms where so far only the existence of algorithms has been shown or the best known algorithms are highly inefficient

the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

the monadic second order logic of graphs ix machines and their behaviours abstract we establish that every monadic second order property of the behaviour of a machine transition systems and tree automata are typical examples of machines is a monadic second order property of the machine itself in this way we clarify the distinction between dynamic properties of machines i e properties of their behaviours and their static properties i e properties of the graphs or relational structures representing them it is important for program verification that the dynamic properties that one wants to verify can be formulated statically in the simplest possible way as a corollary of our main result we also obtain that the monadic theory of an algebraic tree is decidable

grid structures and undecidable constraint theories abstract we prove three new undecidability results for computational mechanisms over finite trees there is a linear ultra shallow noetherian and strongly confluent rewrite system r such that the fragment of the first order theory of one step rewriting by r is undecidable the emptiness problem for tree automata with equality tests between cousins is undecidable and the fragment of the first order theory of set constraints with the union operator is undecidable the common feature of these three computational mechanisms is that they allow us to describe the set of first order terms that represent grids we extend our representation of grids by terms to a representation of linear two dimensional patterns by linear terms which allows us to transfer classical techniques on the grid to terms and thus to obtain our undecidability results

parameterized and approximation complexity of partial vc dimension abstract we introduce the problem partial vc dimension that asks given a hypergraph h x e and integers k and l whether one can select a set c x of k vertices of h such that the set e c e e of distinct hyperedge intersections with c has size at least l the sets e c define equivalence classes over e partial vc dimension is a generalization of vc dimension which corresponds to the case l 2 k and of distinguishing transversal which corresponds to the case l e the latter is also known as test cover in the dual hypergraph we also introduce the associated fixed cardinality maximization problem max partial vc dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices we study the algorithmic complexity of partial vc dimension and max partial vc dimension both on general hypergraphs and on more restricted instances in particular neighborhood hypergraphs of graphs

existential second order logic over graphs charting the tractability frontier fagin s theorem the first important result of descriptive complexity asserts that a property of graphs is in np if and only if it is definable by an existential second order formula in this article we study the complexity of evaluating existential second order formulas that belong to prefix classses of existential second order logic where a prefix class is the collection of all existential second order formulas in prenex normal form such that the second order and the first order quantifiers obey a certain quantifier pattern we completely characterize the computational complexity of prefix classes of existential second order logic in three different contexts 1 over directed graphs 2 over undirected graphs with self loops and 3 over undirected graphs without self loops our main result is that in each of these three contexts a dichotomy holds that is to say each prefix class of existential second order logic either contains sentences that can express np complete problems or each of its sentences expresses a polynomial time solvable problem although the boundary of the dichotomy coincides for the first two cases it changes as one moves to undirected graphs without self loops the key difference is that a certain prefix class based on the well known ackermann class of first order logic contains sentences that can express np complete problems over graphs of the first two types but becomes tractable over undirected graphs without self loops moreover establishing the dichotomy over undirected graphs without self loops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics including results about graphs of bounded tree width and ramsey s theorem

obtaining planarity by contracting few edges the planar contraction problem is to test whether a given graph can be made planar by using at most k edge contractions this problem is known to be np complete we show that it is fixed parameter tractable when parameterized by k

counting modulo quantifiers on finite structures we give a combinatorial method for proving elementary equivalence in first order logic fo with counting modulo n quantifiers dn inexpressibility results for fo dn with built in linear order are also considered for instance the class of linear orders of length divisible by n 1 cannot be expressed in fo dn using this result we prove that comparing cardinalities or connectivity of ordered graphs are not definable in fo dn we also show that the height of complete n ary trees cannot be expressed in fo dn with linear order interpreting the predicate y nx as a complete n ary tree we show that the predicate y px cannot be defined in fo dn with linear order whenever p has a prime factor that does not divide n this solves the problem raised by niwi ski and stolboushkin lics 93 we also discuss a connection between our results and the well known open problem in circuit complexity theory whether acc nc1

induced packing of odd cycles in planar graphs an induced packing of odd cycles in a graph is a packing such that there is no edge in the graph between any two odd cycles in the packing we prove that an induced packing of k odd cycles in an n vertex graph can be found if it exists in time 2 o k 3 2 n 2 e for any constant e 0 when the input graph is planar we also show that deciding if a graph has an induced packing of two odd induced cycles is np complete

finite graph automata for linear and boundary graph languages graph grammars can be regarded as a generalization of context free grammars from strings to graphs over the past 30 years a rich theory of graph grammars and their languages has been developed however there are no graph automata there is no duality between generative and recognizing devices as it is known for the chomsky hierarchy of formal languages here we introduce graph automata as devices for the recognition of sets of undirected node labelled graphs a graph automaton consists of a finite state control a finite set of instructions and a collection of heads or guards it reads an input graph in a systematic way and performs a graph search directed by the instructions as our main results we show that finite graph automata recognize exactly the set of graph languages generated by linear nce graph grammars and that alternating finite graph automata recognize exactly the languages of boundary graph grammars finally we generalize some automata theoretic properties from string to graph automata integrate the connectivity of graphs into graph automata and explain why graph automata cannot be generalized to deal with dynamic edge relabellings and ence graph languages

n by n checkers is exptime complete the game of checkers can easily be generalized to be played on an n by n board and the complexity of deciding questions about positions regarded as a function of n this paper considers mainly the question of whether a particular player can force a win from a given position and also the question of what is the best move in a given position each of these problems is shown to be complete in exponential time this means that any algorithm to solve them must take time which rises exponentially with respect to some power of n and moreover that they are amongst the hardest problems with such a time bound for instance if there are any problems solvable in exponential time but not in polynomial space then these two problems are amongst them

alternation with restrictions on looping alternating turing machines with restrictions preventing them from returning to a previous configuration model games with rules enforcing such a restriction for instance the chinese version of go such restrictions do not affect the time complexity of problems for alternating turing machines but space s on a machine with the restriction is equivalent either to time or to space exponential in s on a normal alternating machine depending on the precise nature of the restriction

towards fixed parameter tractable algorithms for abstract argumentation abstract argumentation frameworks have received a lot of interest in recent years most computational problems in this area are intractable but several tractable fragments have been identified in particular dunne showed that many problems can be solved in linear time for argumentation frameworks of bounded tree width however these tractability results which were obtained via courcelle s theorem do not directly lead to efficient algorithms the goal of this paper is to turn the theoretical tractability results into efficient algorithms and to explore the potential of directed notions of tree width for defining larger tractable fragments as a by product we will sharpen some known complexity results

undirected edge geography abstract the game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex erasing the edge in between the player who first has no legal move is the loser we show that the decision problem of determining whether a position in this game is a win for the first player is pspace complete further the problem remains pspace complete when restricted to planar graphs with maximum degree 3 however if the underlying graph is bipartite we provide 1 a linear algebraic characterization of the p and n positions yielding 2 a polynomial time algorithm for deciding whether any given position is p or n and also 3 a polynomial time algorithm to find winning moves

an algebraic theory of graph reduction we show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules the method is constructive in the sense that we describe an algorithm which will produce from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth k a set of rewrite rules that reduces any member of the class to one of finitely many graphs in a number of steps bounded by the size of the graph this reduction system corresponds to an algorithm that runs in time linear in the size of the graph

size and treewidth bounds for conjunctive queries this article provides new worst case bounds for the size and treewith of the result q d of a conjunctive query q applied to a database d we derive bounds for the result size q d in terms of structural properties of q both in the absence and in the presence of keys and functional dependencies these bounds are based on a novel coloring of the query variables that associates a coloring number c q to each query q intuitively each color used represents some possible entropy of that variable using this coloring number we derive tight bounds for the size of q d in case i no functional dependencies or keys are specified and ii simple functional dependencies keys are given these results generalize recent size bounds for join queries obtained by atserias et al 2008 in the case of arbitrary compound functional dependencies we use tools from information theory to provide lower and upper bounds establishing a close connection between size bounds and a basic question in information theory our new coloring scheme also allows us to precisely characterize both in the absence of keys and with simple keys the treewidth preserving queries the queries for which the treewidth of the output relation is bounded by a function of the treewidth of the input database finally we give some results on the computational complexity of determining the size bounds and of deciding whether the treewidth is preserved

hitting minors on bounded treewidth graphs ii single exponential algorithms abstract for a finite collection of graphs f the f m deletion resp f tm deletion problem consists in given a graph g and an integer k decide whether there exists s v g with s k such that g s does not contain any of the graphs in f as a minor resp topological minor we are interested in the parameterized complexity of both problems when the parameter is the treewidth of g denoted by tw and specifically in the cases where f contains a single connected planar graph h we present algorithms running in time 2 o tw n o 1 called single exponential when h is either p 3 p 4 c 4 the paw the chair and the banner for both h m deletion and h tm deletion and when h k 1 i with i 1 for h tm deletion some of these algorithms use the rank based approach introduced by bodlaender et al 2015 7 this is the second of a series of articles on this topic and the results given here together with other ones allow us in particular to provide a tight dichotomy on the complexity of h m deletion in terms of h

parallel algorithms with optimal speedup for bounded treewidth we describe the first parallel algorithm with optimal speedup for constructing minimum width tree decompositions of graphs of bounded treewidth on n vertex input graphs the algorithm works in o log n 2 time using o n operations on the erew pram we also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a variety of problems on graphs of bounded treewidth including all decision problems expressible in monadic second order logic on n vertex input graphs the algorithms use o n operations together with o log n log n time on the erew pram or o log n time on the crcw pram

existence of multiple periodic solutions for a natural hamiltonian system in a potential well we prove under a pinching hypothesis a theorem of existence of at least n periodic orbits for a closed regular energy hypersurface in r2n which is the level set of a natural i e of the type potential kinetic energy hamiltonian function and projects onto a potential well in the configuration space

parameterized algorithms for conflict free colorings of graphs abstract in this paper we study the conflict free coloring of graphs induced by neighborhoods a coloring of a graph is conflict free if every vertex has a uniquely colored vertex in its neighborhood the conflict free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict free we consider both closed neighborhoods where the neighborhood of a vertex includes itself and open neighborhoods where a vertex does not include in its neighborhood we study the parameterized complexity of conflict free closed neighborhood coloring and conflict free open neighborhood coloring problems we show that both problems are fixed parameter tractable fpt when parameterized by the cluster vertex deletion number of the input graph this generalizes the result of gargano and rescigno theoretical computer science 2015 that conflict free coloring is fpt parameterized by the vertex cover number also we show that both problems admit an additive constant approximation algorithm when parameterized by the distance to threshold graphs we also study the complexity of the problem on special graph classes for split graphs we give a polynomial time algorithm for closed neighborhood conflict free coloring problem whereas we show that open neighborhood conflict free coloring is np complete for cographs we show that conflict free closed neighborhood coloring can be solved in polynomial time and conflict free open neighborhood coloring need at most three colors we show that interval graphs can be conflict free colored using at most four colors

complexity of conflict free colorings of graphs we consider conflict free colorings of graph neighborhoods each vertex of the graph must be assigned a color so that for each vertex v there is at least one color appearing exactly once in the neighborhood of v the goal is to minimize the number of used colors we consider both the case of closed neighborhoods when the neighborhood of a node includes the node itself and the case of open neighborhoods when a node does not belong to its neighborhood in this paper we study complexity aspects of the problem we show that the problem of conflict free coloring of closed neighborhoods is np complete moreover we give non approximability results for the conflict free coloring of open neighborhoods from a positive point of view both problems become tractable if parameterized by the vertex cover number or the neighborhood diversity number of the graph we present simple algorithms which improve on existing results

compatibility of unrooted phylogenetic trees is fpt a collection of t1 t2 tk of unrooted leaf labelled phylogenetic trees all with different leaf sets is said to be compatible if there exists a tree t such that each tree ti can be obtained from t by deleting leaves and contracting edges determining compatibility is np hard and the fastest algorithm to date has worst case complexity of around nk time n being the number of leaves here we present an o nf k algorithm proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable fpt with respect to the number k of trees

the recognizability of sets of graphs is a robust property once the set of finite graphs is equipped with an algebra structure arising from the definition of operations that generalize the concatenation of words one can define the notion of a recognizable set of graphs in terms of finite congruences applications to the construction of efficient algorithms and to the theory of context free sets of graphs follow naturally the class of recognizable sets depends on the signature of graph operations we consider three signatures related respectively to hyperedge replacement hr context free graph grammars to vertex replacement vr context free graph grammars and to modular decompositions of graphs we compare the corresponding classes of recognizable sets we show that they are robust in the sense that many variants of each signature where in particular operations are defined by quantifier free formulas a quite flexible framework yield the same notions of recognizability we prove that for graphs without large complete bipartite subgraphs hr recognizability and vr recognizability coincide the same combinatorial condition equates hr context free and vr context free sets of graphs inasmuch as possible results are formulated in the more general framework of relational structures

the monadic second order logic of graphs xiv uniformly sparse graphs and edge set quantifications we consider the class usk of uniformly k sparse simple graphs i e the class of finite or countable simple graphs every finite subgraph of which has a number of edges bounded by k times the number of vertices we prove that for each k every monadic second order formula intended to express a graph property that uses variables denoting sets of edges can be effectively translated into a monadic second order formula where all set variables denote sets of vertices and that expresses the same property of the graphs in usk this result extends to the class of uniformly k sparse simple hypergraphs of rank at most m for any k and m it follows that every subclass of usk consisting of finite graphs of bounded clique width has bounded tree width clique width is a graph complexity measure similar to tree width and relevant to the construction of polynomial algorithms for np complete problems on special classes of graphs

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

uniform and nonuniform recognizability deterministic and nondeterministic finite state recognizability over finite structures are introduced in an algebraic setting avoiding detailed computational conventions as needed in the definition of finite state acceptors for deterministic recognizability the classical approach is adopted using a uniform homomorphism from the input domain consisting of terms into a finite algebra for the nondeterministic case we refer to relational input structures and to an acceptance via relational homomorphisms which are applied nonuniformly since they depend on the input structures we show how this approach encompasses known models of nondeterministic automata over finite words trees pictures and graphs and present some elementary metaresults connecting uniform recognizability nonuniform recognizability and monadic second order logic

on the path width of integer linear programming we consider the feasibility problem of integer linear programming ilp we show that solutions of any ilp instance can be naturally represented by an fo definable class of graphs for each solution there may be many graphs representing it however one of these graphs is of path width at most 2n where n is the number of variables in the instance since fo is decidable on graphs of bounded path width we obtain an alternative decidability result for ilp the technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage we also show how this new result links to automata theory and program verification

the monadic second order logic of graphs x linear orderings graphs are finite and handled as relational structures we give some answers to the following general questions 1 1 for which classes of graphs c is it possible to specify a linear ordering of the set of vertices of each graph of c by fixed monadic second order formulas 2 2 for which classes of graphs c does there exist an extension l of monadic second order logic such that a subclass l of c is recognizable if and only if it is the class of graphs in c that satisfy a formula of l in this paper recognizability is understood in an algebraic sense relative to a finite set of graph operations and basic graphs that generate all graphs of c 3 3 for which classes of graphs c is it possible to construct in every graph of the class and by fixed formulas of a suitable extension of monadic second order logic its hierarchichal structure i e a finite term written with the operations and basic graphs of 2 that defines the considered graph applications concern dependency graphs of partially commutative words partial k paths cographs and graphs the modular decomposition of which uses prime graphs of bounded size

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

increasing the minimum degree of a graph by contractions abstract the degree contractibility problem is to test whether a given graph g can be modified to a graph of minimum degree at least d by using at most k contractions we prove the following three results first degree contractibility is np complete even when d 14 second it is fixed parameter tractable when parameterized by k and d third it is w 1 hard when parameterized by k we also study its variant where the input graph is weighted i e has some edge weighting and the contractions preserve these weights the weighted degree contractibility problem is to test if a weighted graph g can be contracted to a weighted graph of minimum weighted degree at least d by using at most k weighted contractions we show that this problem is np complete and that it is fixed parameter tractable when parameterized by k in addition we pinpoint a relationship with the problem of finding a minimal edge cut of maximum size in a graph and study the parameterized complexity of this problem and its variants

complexity of path forming games abstract for a number of two player games where players alternately choose the next vertex of a simple or an elementary path in a graph we consider the problem to determine whether for a given game instance there is a winning strategy for the first player we show several of these problems to be pspace complete in some special cases we obtain polynomial time algorithms based on graph rewriting or an intricate form of dynamic programming e g we show generalized geography and some other pspace complete problems to be linear time solvable on graphs with constant bounded treewidth

hierarchically specified unit disk graphs we characterize the complexity of several basic optimization problems for unit disk graphs specified hierarchically as in lw87a le88 lw92 both pspace hardness results and polynomial time approximations are presented for most of the problems considered these problems include minimum vertex coloring maximum independent set minimum clique cover minimum dominating set and minimum independent dominating set

a parametrized algorithm for matroid branch width branch width is a structural parameter very closely related to tree width but branch width has an immediate generalization from graphs to matroids we present an algorithm that for a given matroid m of bounded branch width t which is represented over a finite field finds a branch decomposition of m of width at most 3t in cubic time then we show that the branch width of m is a uniformly fixed parameter tractable problem other applications include recognition of matroid properties definable in the monadic second order logic for bounded branch width and s i oum approximating rank width and clique width quickly in proceedings of the 31st international workshop on graph theoretic concepts in computer science springer verlag heidelberg to appear a cubic time approximation algorithm for graph rank width and clique width

existence of multiple periodic orbits of hamiltonian systems on positive type hypersurfaces in r2n abstract this paper deals with the fixed energy problem of hamiltonian systems in r 2n the main result is an existence theorem of multiple periodic orbits for hamiltonian vector fields on a class of contact type but not necessarily star shaped hypersurfaces this is the first multiplicity result for hypersurfaces that are more general than star shaped ones the main result is achieved by using critical point theory for a special hamiltonian function and central gravity of the proof relies on a delicate construction of such hamiltonian function

on the decidability of certain integer subgraph problems on context free graph languages we show the decidability of integer subgraph problems isps on context free sets of graphs l defined by hyperedge replacement systems hrss additionally we give a very general characterization of isps to be decidable on a set l an isp consists of a property s and a mapping f if j is a subgraph of a graph g then s g j is true or false and f g j is an integer we show the decidability of the following problem let 1 n be n isps that fulfill our characterization and let c be a set of conditions i o j that specify two isps i and j and a compare symbol o given a context free set of graphs l defined by a hrs is there a graph g l that has n subgraphs j1 j n such that s prod i g j holds true for i 1 n and s prod i g j i o s prod j g j j for each condition i o j

all structured programs have small tree width and good register allocation the register allocation problem for an imperative program is often modelled as the coloring problem of the interference graph of the control flow graph of the program the interference graph of a flow graph g is the intersection graph of some connected subgraphs of g these connected subgraphs represent the lives or life times of variables so the coloring problem models that two variables with overlapping life times should be in different registers for general programs with unrestricted gotos the interference graph can be any graph and hence we cannot in general color within a factor o n from optimality unless np p

the role of planarity in connectivity problems parameterized by treewidth for some years it was believed that for connectivity problems such as hamiltonian cycle algorithms running in time 2 o tw n o 1 called single exponential existed only on planar and other topologically constrained graph classes where tw stands for the treewidth of the n vertex input graph this was recently disproved by cygan et al 3 focs 2011 bodlaender et al 1 icalp 2013 and fomin et al 11 soda 2014 who provided single exponential algorithms on general graphs for most connectivity problems that were known to be solvable in single exponential time on topologically constrained graphs in this article we further investigate the role of planarity in connectivity problems parameterized by treewidth and convey that several problems can indeed be distinguished according to their behavior on planar graphs known results from the literature imply that there exist problems like cycle packing that cannot be solved in time 2 o tw log tw n o 1 on general graphs but that can be solved in time 2 o tw n o 1 when restricted to planar graphs our main contribution is to show that there exist natural problems that can be solved in time 2 o tw log tw n o 1 on general graphs but that cannot be solved in time 2 o tw log tw n o 1 even when restricted to planar graphs furthermore we prove that planar cycle packing and planar disjoint pathscannot be solved in time 2 o tw n o 1 the mentioned negative results hold unless the eth fails we feel that our results constitute a first step in a subject that can be further exploited

the monadic second order logic of graphs vii graphs as relational structures abstract relational structures form a unique framework in which various types of graphs and hypergraphs can be formalized and studied we define operations on structures that are compatible with monadic second order logic and that are powerful enough to represent context free graph and hypergraph grammars of various types namely the so called hyperedge replacement c ednce and separated handle rewriting grammars several results concerning monadic second order properties of the generated sets are obtained in a uniform way we also give a logical characterization of the equational sets of structures that generalizes the ones obtained by engelfriet and courcelle for hyperedge replacement and c ednce sets of graphs

an extension of the bianchi egnell stability estimate to bakry gentil and ledoux s generalization of the sobolev inequality to continuous dimensions abstract this paper extends a stability estimate of the sobolev inequality established by bianchi and egnell in 3 bianchi and egnell s stability estimate answers the question raised by h brezis and e h lieb in 5 is there a natural way to bound 2 2 c n 2 2 n n 2 2 from below in terms of the distance of from the manifold of optimizers in the sobolev inequality establishing stability estimates also known as quantitative versions of sharp inequalities of other forms of the sobolev inequality as well as other inequalities is an active topic see 9 11 and 12 for stability estimates involving sobolev inequalities and 6 11 and 14 for stability estimates on other inequalities in this paper we extend bianchi and egnell s stability estimate to a sobolev inequality for continuous dimensions bakry gentil and ledoux have recently proved a sharp extension of the sobolev inequality for functions on r r n which can be considered as an extension to continuous dimensions v h nguyen determined all cases of equality the present paper extends the bianchi egnell stability analysis for the sobolev inequality to this continuous dimensional generalization

meta kernelization in a parameterized problem every instance i comes with a positive integer k the problem is said to admit a polynomial kernel if in polynomial time one can reduce the size of the instance i to a polynomial in k while preserving the answer in this work we give two meta theorems on kernelization the first theorem says that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus these theorems unify and extend all previously known kernelization results for planar graph problems

syntactic recognizability of graphs with fuzzy attributes abstract we present a syntactic recognition theory for graphs with fuzzy edges and fuzzy nodes by employing the algebraic structure of magmoids several examples are illustrated and it is proved that the cut of a recognizable set of fuzzy graphs is not always recognizable as a set of crisp graphs the described theory is then applied for the recognition of fuzzy cognitive maps and the recognizability of interesting examples of sets of fuzzy cognitive maps is examined

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

local search is a ptas for feedback vertex set in minor free graphs abstract we show that an extremely simple local search gives a ptas for the feedback vertex set fvs problem in minor free graphs it keeps exchanging a constant number of vertices to improve the current solution until a local optimum is reached the previous ptas by fomin lokshtanov raman and saurabh 1 despite theoretical efficiency is much more complicated in the sense that it combines many advanced algorithmic tools such as contraction decomposition framework by demaine and hajiaghayi 2 courcelle s theorem 3 and the robertson and seymour decomposition 4 our main technical contribution is to show that the local optimum only differs the global optimum by a 1 factor we do so by introducing steiner points those who are not in the local and optimal solutions to the standard analysis of local search we believe that our technique is of independent interest

hierarchical planarity testing algorithms using hierarchical definitions one can describe very large graphs in small space the blow up from the length of the hierarchical description to the size of the graph can be as large as exponential if the efficiency of graph algorithms is measured in terms of the length of the hierarchical description rather than in terms of the graph size algorithms that do not exploit the hierarchy become hopelessly inefficient whether the hierarchy can be exploited to speed up the solution of graph problems depends on the hierarchical graph model in the literature hierarchical graph models have been described that allow almost no exploitation of the hierarchy 16 in this paper a hierarchical graph model that permits taking advantage of the hierarchy is presented for this model algorithms are given that test planarity of a hierarchically described graph in linear time in the length of the hierarchical description

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

a linear time algorithm for metric dimension of cactus block graphs an undirected graph g v e has metric dimension at most k if there is a vertex set u v such that u k and u v v u v there is a vertex w u such that d g w u d g w v where d g u v is the distance the length of a shortest path in an unweighted graph between u and v the metric dimension of g is the smallest integer k such that g has metric dimension at most k a cactus block graph is an undirected graph whose biconnected components are either cycles or complete graphs we present a linear time algorithm for computing the metric dimension of cactus block graphs

an axiomatic definition of context free rewriting and its application to nlc graph grammars abstract an abstract notion of context free grammar is introduced it deals with abstract objects that can be words trees graphs or other combinatorial objects it is applied to nlc graph grammars introduced by rozenberg and janssens the monadic second order theory of a context free nlc set of graphs is decidable

efficient parallel algorithms for parameterized problems abstract a parameterized problem is fixed parameter parallelizable fpp if it can be solved in o f k log n time using o g k n processors where n is the input size k is the parameter f and g are arbitrary computable functions and are constants independent of n and k we re examine the k vertex cover problem from a parameterized parallel complexity standpoint and present a parallel algorithm that outperforms the previous known algorithm using o m instead of o n 2 processors the running time improves from o k k to o k 3 log n 1 2738 k where n and m are the number of vertices and edges of the input graph respectively this is achieved by first showing that vertex cover kernelization that is based on crown decomposition is in fpp as well finally we consider the use of the recently introduced modular width parameter in particular we show that the weighted maximum clique problem is fpp when parameterized by this auxiliary parameter

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

algorithms for graph problems on bnlc structured graphs abstract we present algorithms for analyzing single graphs and sets of graphs generated by boundary node label controlled bnlc graph grammars this paper extends a unified framework for developing algorithms on context free graph languages to bnlc graph languages graphs in bnlc graph languages do not necessarily have vertex separators of bounded size as do graphs in context free graph languages we give combinatorial decision and query algorithms on single graphs generated by a deterministic bnlc graph grammar and algorithms for the following question does the language of a given bnlc graph grammar contain a graph that fulfills a certain graph property all algorithms developed in this paper consider the graph grammar as input

the art of trellis decoding is fixed parameter tractable given n subspaces of a finite dimensional vector space over a fixed finite field mathbb f we wish to find a linear layout v 1 v 2 ldots v n of the subspaces such that dim v 1 v 2 cdots v i cap v i 1 cdots v n le k for all i such a linear layout is said to have width at most k when restricted to 1 dimensional subspaces this problem is equivalent to computing the trellis width or minimum trellis state complexity of a linear code in coding theory and computing the path width of an mathbb f represented matroid in matroid theory we present a fixed parameter tractable algorithm to construct a linear layout of width at most k if it exists for input subspaces of a finite dimensional vector space over mathbb f as corollaries we obtain a fixed parameter tractable algorithm to produce a path decomposition of width at most k for an input mathbb f represented matroid of path width at most k and a fixed parameter tractable algorithm to find a linear rank decomposition of width at most k for an input graph of linear rank width at most k in both corollaries no such algorithms were known previously our approach is based on dynamic programming combined with the idea developed by bodlaender and kloks 1996 for their work on path width and tree width of graphs it was previously known that a fixed parameter tractable algorithm exists for the decision version of the problem for matroid path width a theorem by geelen gerards and whittle 2002 implies that for each fixed finite field mathbb f there are finitely many forbidden mathbb f representable minors for the class of matroids of path width at most k an algorithm by hlin n 2006 can detect a minor in an input mathbb f represented matroid of bounded branch width however this indirect approach would not produce an actual path decomposition our algorithm is the first one to construct such a path decomposition and does not depend on the finiteness of forbidden minors

efficient solution to connectivity problems on hierarchically defined graphs using hierarchical definitions we can describe very large graphs in small spaces in this paper we discuss how such succinct graph descriptions can be used to speed up the solution of graph problems we present the bottom up method that solves graph problems without expanding the hierarchical description this allows solutions that are efficient in terms of the hierarchical graph description instead of the size of the expanded graph we exemplify the method by giving efficient solutions to connectivity problems on hierarchical graphs our results have applications in computer aided design for integrated circuit design and other engineering problems

the monadic second order logic of graphs xii planar graphs and planar maps we prove that we can specify by formulas of monadic second order logic the unique planar embedding of a 3 connected planar graph if the planar graph is not 3 connected but given with a linear order of its set of edges we can also define a planar embedding by monadic second order formulas we cannot do so in general without the ordering even for 2 connected planar graphs the planar embedding of a graph can be specified by a relational structure called a map which is a graph enriched with a circular ordering of the edges incident with each vertex this circular ordering called a rotation system represents a planar embedding of the neighbourhood of each vertex for each connected map one can define a linear order on its vertices by formulas of monadic second order logic hence we have for planar graphs some kind of equivalence between linear orderings and planar embeddings

boundary classes for graph problems involving non local properties abstract we continue the study of boundary classes for np hard problems and focus on seven np hard graph problems involving non local properties hamiltonian cycle hamiltonian cycle through specified edge hamiltonian path feedback vertex set connected vertex cover connected dominating set and graph vc con dimension our main result is the determination of the first boundary class for feedback vertex set we also determine boundary classes for hamiltonian cycle through specified edge and hamiltonian path and give some insights on the structure of some boundary classes for the remaining problems

new limits of treewidth based tractability in optimization sparse structures are frequently sought when pursuing tractability in optimization problems they are exploited from both theoretical and computational perspectives to handle complex problems that become manageable when sparsity is present an example of this type of structure is given by treewidth a graph theoretical parameter that measures how tree like a graph is this parameter has been used for decades for analyzing the complexity of various optimization problems and for obtaining tractable algorithms for problems where this parameter is bounded the goal of this work is to contribute to the understanding of the limits of the treewidth based tractability in optimization our results are as follows first we prove that in a certain sense the already known positive results on extension complexity based on low treewidth are the best possible secondly under mild assumptions we prove that treewidth is the only graph theoretical parameter that yields tractability for a wide class of optimization problems a fact well known in graphical models in machine learning and in constraint satisfaction problems which here we extend to an approximation setting in optimization

periodic solutions on hypersurfaces and a result by c viterbo on considere un champ vectoriel hamiltonien x j h x x h x sur x r 2n h etant une fonction lisse dont le gradient h est defini par rapport a la metrique euclidienne on cherche des solutions periodiques sur une surface d energie donnee s x r 2n h x const supposee reguliere

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

kernel bounds for path and cycle problems connectivity problems like k path and k disjoint paths relate to many important milestones in parameterized complexity namely the graph minors project color coding and the recent development of techniques for obtaining kernelization lower bounds this work explores the existence of polynomial kernels for various path and cycle problems by considering nonstandard parameterizations we show polynomial kernels when the parameters are a given vertex cover a modulator to a cluster graph or a promised max leaf number we obtain lower bounds via cross composition e g for hamiltonian cycle and related problems when parameterized by a modulator to an outerplanar graph

stability for a gns inequality and the log hls inequality with application to the critical mass keller segel equation starting from the quantitative stability result of bianchi and egnell for the 2 sobolev inequality we deduce several dierent stability results for a gagliardo nirenberg sobolev inequality in the plane then exploiting the connection between this inequality and a fast diusion equation we get stability for the log hls inequality finally using all these estimates we prove a quantitative convergence result for the critical mass keller segel system

sharp stability theorems for the anisotropic sobolev and log sobolev inequalities on functions of bounded variation abstract combining rearrangement techniques with gromov s proof via optimal mass transportation of the 1 sobolev inequality we prove a sharp quantitative version of the anisotropic sobolev inequality on b v r n we also deduce as a corollary of this result a sharp stability estimate for the anisotropic 1 log sobolev inequality

on the parameterized complexity of recognizing well covered r l graph an r ell partition of a graph g is a partition of its vertex set into r independent sets and ell cliques a graph is r ell if it admits an r ell partition a graph is well covered if every maximal independent set is also maximum a graph is r ell well covered if it is both r ell and well covered in this paper we consider two different decision problems in the r ell well covered graph problem r ell wcg for short we are given a graph g and the question is whether g is an r ell well covered graph in the well covered r ell graph problem wc r ell g for short we are given an r ell graph g together with an r ell partition of v g into r independent sets and ell cliques and the question is whether g is well covered we classify most of these problems into p conp complete np complete np hard or conp hard only the cases wc r 0 g for r ge 3 remain open in addition we consider the parameterized complexity of these problems for several choices of parameters such as the size alpha of a maximum independent set of the input graph its neighborhood diversity or the number ell of cliques in an r ell partition in particular we show that the parameterized problem of deciding whether a general graph is well covered parameterized by alpha can be reduced to the wc 0 ell g problem parameterized by ell and we prove that this latter problem is in xp but does not admit polynomial kernels unless mathsf conp subseteq mathsf np mathsf poly

games on interval and permutation graph representations we describe combinatorial games on graphs in which two players antagonistically build a representation of a subgraph of a given graph we show that for a large class of these games determining whether a given instance is a winning position for the next player is pspace hard in contrast we give polynomial time algorithms for solving some versions of the games on trees

testing first order properties for subclasses of sparse graphs we present a linear time algorithm for deciding first order fo properties in classes of graphs with bounded expansion a notion recently introduced by neset il and ossona de mendez this generalizes several results from the literature because many natural classes of graphs have bounded expansion graphs of bounded tree width all proper minor closed classes of graphs graphs of bounded degree graphs with no subgraph isomorphic to a subdivision of a fixed graph and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges we deduce that there is an almost linear time algorithm for deciding fo properties in classes of graphs with locally bounded expansion more generally we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion after a linear time initialization the data structure allows us to test an fo property in constant time and the data structure can be updated in constant time after addition deletion of an edge provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class all our results also hold for relational structures and are based on the seminal result of neset il and ossona de mendez on the existence of low tree depth colorings

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious