Super-$\lambda$ Connectivity of Bipartite Graphs

Abstract

Let $G=(V,E)$ be a connected graph. $G$ is $super-\lambda$ if every minimum edge cut of $G$ isolates a vertex. Moreover, an edge set $S\subseteq E$ is a $restrictededgecut$ of $G$ if $G–S$ is disconnected and every component of $G–S$ has at least $2$ vertices. The $restrictededgeconnectivity$ of $G$, denoted by ${\lambda }^{\prime }(G)$, is the minimum cardinality of all restricted edge cuts. Let $\xi (G)=min\{d_G(u)+d_G(v)–2:uv\in E(G)\}$. We say $G$ is ${\lambda }^{\prime }-optimal$ if ${\lambda }^{\prime }(G)=\xi (G)$. In this paper, we provide a sufficient condition for bipartite graphs to be both super-$\lambda$ and ${\lambda }^{\prime }$-optimal.