Parallel algorithms for finding the most vital edge with respect to minimum spanning tree

Abstract

Given a weighted graph G, the weight of a spanning tree T, denoted by w( T), is defined as the total weight of all edges in T. A spanning tree T in G is called a minimum spanning tree if w( T)⩽ w( T′) for all spanning trees T′ in G. Let w( G) denote the weight of the minimum spanning tree of G if G is connected; otherwise, w( G) = ∞. An edge e is called a most vital edge in G if w( G− e) ⩾ w( G− e′) for every edge e′ of G where G− e′ denotes the partial graph obtained by removing e′ from G. In this paper, we present several cost-optimal parallel algorithms, under different computation models, to find the most vital edge in a weighted graph.