Condensation (graph theory)

In the mathematical theory of directed graphs, a graph is said to be strongly connected or diconnected if every vertex is reachable from every other vertex. The strongly connected components or diconnected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time.

A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them.

The binary relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components. Equivalently, a strongly connected component of a directed graph G is a subgraph that is strongly connected, and is maximal with this property: no additional edges or vertices from G can be included in the subgraph without breaking its property of being strongly connected. The collection of strongly connected components forms a partition of the set of vertices of G.

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 strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every nontrivial strongly connected component contains at least one directed cycle.

Several algorithms compute strongly connected components in linear time.

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