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 \({restricted\; edge\; cut}\) of \(G\) if \(G – S\) is disconnected and every component of \(G – S\) has at least \(2\) vertices. The \({restricted \;edge\; connectivity}\) of \(G\), denoted by \(\lambda'(G)\), 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 \({\lambda’-optimal}\) if \(\lambda'(G) = \xi(G)\). In this paper, we provide a sufficient condition for bipartite graphs to be both super-\(\lambda\) and \(\lambda’\)-optimal.