Vulnerability of super extra edge-connected graphs

Edge connectivity is a crucial measure of the robustness of a network. Several edge connectivity variants have been proposed for measuring the reliability and fault tolerance of networks under various conditions. Let G be a connected graph, S be a subset of edges in G, and k be a positive integer. If G - S is disconnected and every component has at least k vertices, then S is a k-extra edge-cut of G. The k-extra edge-connectivity, denoted by lambda(k)(G), is the minimum cardinality over all k-extra edge-cuts of G. If lambda(k)(G) exists and at least one component of G - S contains exactly k vertices for any minimum k-extra edge-cut S, then G is super-lambda(k). Moreover, when G is super-lambda(k), the persistence of G, denoted by rho(k)(G), is the maximum integer m for which G - F is still super-lambda(k) for any set F subset of E (G) with vertical bar F vertical bar <= m. Previously, bounds of rho(k)(G) were provided only for k is an element of {1,2}. This study provides the bounds of rho(k)(G) for k >= 2. (C) 2019 Elsevier Inc. All rights reserved.