Strongly connected graph

I f each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G. A directed graph is acyclic if and only if it has no connected and every nontrivial strongly connected component contains at least one directed cycle.

Algorithms

• Kosaraju's algorithm uses two passes of transpose graph of the original graph, and each recursive exploration finds a single new strongly connected described it (but did not publish his results) in 1978; Micha Sharir later published it in 1981.
• Tarjan's strongly connected components 1972, performs a single pass of depth first search. It maintains a
• The path-based strong component algorithm uses a to keep track of the vertices not yet assigned to components, while the other kee p s track of the published by Edsger W. Dijkstra in 1976.

Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm are favoured in practice since they require only o

Applications

Algorithm s for finding strongly connected components may be used to variables wi th constraints on the values of pairs of variables): as showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both c ontained in

Strongly connected components are also used to compute the Dulmage–Mendelsohn decomposition, a of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching

Related results

A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path

According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is

See also

• Connected component
• Modular decomposition

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