Construction of Simple Path Graphs in Transport Networks: II. Analysis of Graphs’ Biconnectivity

The problem of constructing all simple paths in an undirected graph that pairwise connect vertices from the given set is interpreted as constructing a graph that is a union of these paths. This construction relies on the computation of biconnected components and bridges of the source graph. A new algorithm which makes this computation more transparent and controllable is proposed. Its idea is to track opening and closing of chords at the depth-first search. Due to the clarity of the proposed solution, its correctness is obvious without much justification. This is its advantage over the well-known Hopcroft–Tarjan algorithm. The proposed algorithm makes self-evident the obtaining biconnected components of a graph by successive uniting its fundamental cycles that have common edges.