Vulnerability of super extra edge-connected graphs

Abstract 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 λ k ( G ) , is the minimum cardinality over all k-extra edge-cuts of G. If λ 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- λ k . Moreover, when G is super- λ k , the persistence of G, denoted by ρ k ( G ) , is the maximum integer m for which G − F is still super- λ k for any set F ⊆ E ( G ) with | F | ≤ m . Previously, bounds of ρ k ( G ) were provided only for k ∈ { 1 , 2 } . This study provides the bounds of ρ k ( G ) for k ≥ 2 .