Vulnerability of super edge-connected graphs

Abstract

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 $\la^{(h)}(G)$ of $G$ is defined as the minimum cardinality over all $h$-extra edge-cuts of $G$. A graph $G$, if $\la^{(h)}(G)$ exists, is super-$\la^{(h)}$ if every minimum $h$-extra edge-cut of $G$ isolates at least one connected subgraph of order $h+1$. The persistence ${\rho }^{(h)}(G)$ of a super-$\la^{(h)}$ graph $G$ is the maximum integer $m$ for which $G-F$ is still super-$\la^{(h)}$ for any set $F\subseteq E(G)$ with $|F|⩽m$. Hong {\it et al.} [Discrete Appl. Math. 160 (2012), 579-587] showed that $\min\{\la^{(1)}(G)-δ(G)-1,δ(G)-1\}\leqslant ρ^{(0)}(G)\leqslant δ(G)-1$, where $\delta (G)$ is the minimum vertex-degree of $G$. This paper shows that $\min\{\la^{(2)}(G)-ξ(G)-1,δ(G)-1\}\leqslant ρ^{(1)}(G)\leqslant δ(G)-1$, where $\xi (G)$ is the minimum edge-degree of $G$. In particular, for a $k$-regular super-$\la'$ graph $G$, ${\rho }^{(1)}(G)=k-1$ if $\la^{(2)}(G)$ does not exist or $G$ is super-$\la^{(2)}$ and triangle-free, from which the exact values of ${\rho }^{(1)}(G)$ are determined for some well-known networks.