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doi:10.1016/S0020-0190(98)00077-5 
Copyright © 1998 Published by Elsevier Science B.V.
Finding the detour-critical edge of a shortest path between two nodes*1
Received 15 August 1997;
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Abstract
Let PG(r, s) denote a shortest path between two nodes r and s in an undirected graph G with nonnegative edge weights. A detour at a node u ε PG(r, s) = 
r,…, u, v,…,s
is defined as a shortest path PG − e(u, s) from u to s which does not make use of (u, v). In this paper we focus on the problem of finding an edge e = (u, v) ε PG(r, s) whose removal produces a detour at node u such that the length of PG − e(u, s) minus the length of PG(u, s) is maximum. We call such an edge a detour-critical edge. We will show that this problem can be solved in O(m + n log n) time, where n and m denote the number of nodes and edges in the graph, respectively.
Author Keywords: Shortest path; Fault tolerance; Transient edge failures; Longest detour; Most critical edge
