Increasing the Edge-Connectivity by Contracting a Vertex Subset in Graphs

Let G = (V,E) be an edge weighted graph with n vertices and m edges. For a given integer p with 1 <p <n, we call a set X ⊆ V of p vertices a p-maximizer if X has a property that the edge-connectivity of the graph obtained by contracting X into a single vertex is no less than that of the graph obtained by contracting any other subset of p vertices. In this paper, we first show that there always exists an ordering v1,v2,...,vn of vertices in V such that, for each i = 2,3,...,n - 1, set {v1,v2,...,vi} is an i-maximizer. We give an O(mn + n2log n) time algorithm for finding such an ordering and then show an application to the source location problem.