A faster computation of the most vital edge of a shortest path

Abstract Let P G (r,s) denote a shortest path between two nodes r and s in an undirected graph G=(V,E) such that |V|=n and |E|=m and with a positive real length w(e) associated with any e∈E . In this paper we focus on the problem of finding an edge e ∗ ∈P G (r,s) whose removal is such that the length of P G−e ∗ (r,s) is maximum, where G−e ∗ =(V,E⧹{e ∗ }) . Such an edge is known as the most vital edge of the path P G (r,s) . We will show that this problem can be solved in O (m·α(m,n)) time, where α is the functional inverse of the Ackermann function, thus improving on the previous O (m+n log n) time bound.