Tarjan's strongly connected components algorithm: Difference between revisions

Revision as of 20:54, 1 August 2025

Revision as of 00:45, 22 August 2025

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== Overview ==

The algorithm takes a [[directed graph]] as input, and produces a [[Partition of a set|partition]] of the graph's [[Vertex (graph theory)|vertices]] into the graph's strongly connected components. Each vertex of the graph appears in exactly one of the strongly connected components. Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, aany vertex whose in-degree or out-degree is 0, or anyevery vertex of ana [[directed acyclic graph]].

The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on any nodes that have not yet been found). As usual with depth-first search, the search visits every node of the graph exactly once, refusing to revisit any node that has already been visited. Thus, the collection of search trees is a [[Spanning forest#Spanning forests|spanning forest]] of the graph. The strongly connected components will be recovered as certain subtrees of this forest. The roots of these subtrees are called the "roots" of the strongly connected components. Any node of a strongly connected component might serve as a root, if it happens to be the first node of a component that is discovered by search.