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.
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.
directed acyclic graph
is the condensation of the blue directed graph. It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex.
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.