Finding the Anti-block Vital Edge of a Shortest Path Between Two Nodes

Let PG(s, t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u ∈ PG(s, t) = (s, . . . , u, v, . . . , t) is defined as a shortest path PG−e(u, t) from u to t which does not make use of (u, v). In this paper, we focus on the problem of finding an edge e = (u, v) ∈ PG(s, t) whose removal produces a detour at node u such that the ratio of the length of PG−e(u, t) to the length of PG(u, t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown.