Vulnerability of super edge-connected graphs

A subset F of edges in a connected graph G is a h-extra edge-cut if G − F is disconnected and every component has more than h vertices. The h-extra edge-connectivity λ(G) of G is defined as the minimum cardinality over all h-extra edge-cuts of G. A graph G, if λ(G) exists, is super-λ if every minimum h-extra edge-cut of G isolates at least one connected subgraph of order h + 1. The persistence ρ(G) of a super-λ graph G is the maximum integer m for which G − F is still super-λ for any set F ⊆ E(G) with |F | 6 m. Hong et al. [Discrete Appl. Math. 160 (2012), 579-587] showed that min{λ(G) − δ(G) − 1, δ(G) − 1} 6 ρ(G) 6 δ(G) − 1, where δ(G) is the minimum vertex-degree of G. This paper shows that min{λ(G) − ξ(G) − 1, δ(G)− 1} 6 ρ(G) 6 δ(G)− 1, where ξ(G) is the minimum edge-degree of G. In particular, for a k-regular super-λ graph G, ρ(G) = k − 1 if λ(G) does not exist or G is super-λ and triangle-free, from which the exact values of ρ(G) are determined for some well-known networks.