Path-based depth-first search for strong and biconnected components

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      For a directed graph G = (V, E), we define an SCC as a subset Vs of V, where for every X ∈ Vs and Y ∈ V, if there exists a path from X to Y and a path from Y to X, then Y ∈ Vs. A path is a sequence of vertices, where if X and Y are two consecutive vertices in the path, then (X, Y) ∈ E. In the research literature, many algorithms have been proposed for the SCC-finding problem requiring linear time and based on the DFS algorithm (Gabow, 2000; Sharir, 1981). However, Tarjan’s algorithm (Tarjan, 1972) is considered to be one of the most efficient and requires a linear search time on the order of O(v + e), where v is the number of vertices and e is the number of edges in the input graph.

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      Discovering SCCs knowledge of digraphs is one of crucial graph problems. The existing methods for finding SCCs are designed mostly based on two graph traversal algorithms, depth-first search (DFS) [11–13] and breadth-first search (BFS) [14,15]. However, traditional graph theory approaches focus on solving deterministic problem [16].

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