Bridge (graph theory)
In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge can uniquely determine a cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges. This type of bridge should be distinguished from an unrelated meaning of "bridge" in graph theory, a subgraph separated from the rest of the graph by a  specified subset of vertices; see . Trees and forests. A graph with formula_1 nodes can contain at most formula_2 bridges, since adding additional edges must create a cycle. The graphs with exactly formula_2 bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests. In every undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint paths connecting them. (Every vertex is related to itself via two length-zero paths, which are identical but nevertheless edge-disjoint .) The equivalence classes of this relation are called 2-edge-connected components, and the bridges of the graph are exactly the edges whose endpoints belong to different components. The bridge-block tree of the graph has a vertex for every nontrivial component and an edge for every bridge. Relation to vertex connectivity. Bridges are closely related to the concept of articulation vertices, vertices that belong to every path between some pair of other vertices. The two endpoints of a bridge are articulation vertices unless they have a degree of 1, although it may also be possible for a non-bridge  edge to have two articulation vertices as endpoints. Analogously to bridgeless graphs being 2-edge-connected, graphs without articulation vertices are 2-vertex-connected. In a cubic graph, every cut vertex is an endpoint of at least one bridge. Bridgeless graphs. A bridgeless graph is a graph that does not have any bridges. Equivalent conditions are that each connected component of the graph has an open ear decomposition, that each connected component is 2-edge-connected, or (by Robbins' theorem) that every connected component has a strong orientation. An important open problem involving bridges is the cycle double cover conjecture , due to Seymour and Szekeres (1978 and 1979, independently), which states that every bridgeless graph admits a multi-set of simple cycles which contains each edge exactly twice. Tarjan's bridge-finding algorithm. The first linear time algorithm for finding the bridges in a graph was described by Robert Tarjan in 1974. It performs the following steps: Bridge-finding with chain decompositions. A very simple bridge-finding algorithm uses chain decompositions. Chain decompositions do not only allow to compute all bridges of a graph, they also allow to "read off" every cut vertex of "G" (and the block-cut tree  of "G"), giving a general framework for testing 2-edge- and 2-vertex-connectivity (which extends to linear-time 3-edge- and 3-vertex-connectivity tests). Chain decompositions are special ear decompositions depending on a DFS-tree "T" of "G" and can be computed very simply: Let every vertex be marked as unvisited. For each vertex "v" in ascending DFS-numbers 1..."n", traverse every backedge (i.e. every edge not in the DFS tree) that is incident to "v" and follow the path of tree-edges back to the root of "T", stopping at the first vertex that is marked  as visited. During such a traversal, every traversed vertex is marked as visited. Thus, a traversal stops at the latest at "v" and forms either a directed path or cycle, beginning with v; we call this path or cycle a "chain". The "i"th chain found by this procedure is referred to as "C". "C=C,C..." is then a "chain decomposition" of "G". The following characterizations then allow to "read off" several properties of "G" from "C" efficiently, including all bridges of "G". Let "C" be a chain decomposition of a simple connected graph "G=(V,E)". Bridge (graph theory