Bounds for the Super Extra Edge Connectivity of Graphs

Abstract

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 $F\subseteq E(G)$ with $|F|\le m$. It has been shown that the bounds of $\rho _k(G)$ when $k\in \{1,2\}$. This study shows the bounds of $\rho _k(G)$ when $k\ge 3$.