Strongly connected component

Graph with strongly connected components marked

A directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex. In particular, this means paths in each direction; a path from a to b and also a path from b to a.

The strongly connected components of a directed graph G are its maximal strongly connected subgraphs. If 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 (nontrivial) strongly connected subgraphs (because a cycle is strongly connected, and every strongly connected graph contains at least one cycle).

Kosaraju's algorithm, Tarjan's algorithm and Gabow's algorithm all efficiently compute the strongly connected components of a directed graph, but Tarjan's and Gabow's are favoured in practice since they require only one depth-first search rather than two.

Algorithms for finding strongly connected components may be used to solve 2-satisfiability problems (systems of Boolean variables with constraints on the values of pairs of variables): as Aspvall, Plass & Tarjan (1979) showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both contained in the same strongly connected component of the implication graph of the instance.

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 2-edge-connected.

See also

• Connected component

References

• Aspvall, Bengt; Plass, Michael F.; Tarjan, Robert E. (1979), "A linear-time algorithm for testing the truth of certain quantified boolean formulas", Information Processing Letters 8 (3): 121–123, doi:10.1016/0020-0190(79)90002-4 .
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 22.5, pp. 552–557.