Finding the anti-block vital edge of a shortest path between two nodes

Let P G ( 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 ∈ P G ( s , t )=( s ,…, u , v ,…, t ) is defined as a shortest path P G − 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 )∈ P G ( s , t ) whose removal produces a detour at node u such that the ratio of the length of P G − e ( u , t ) to the length of P G ( 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.