On the Connectivity and Superconnectivity of Bipartite Digraphs and Graphs

Abstract

In this work, first, we present sufficient conditions for a bipartite digraph to attain optimum values of a stronger measure of connectivity, the so-called superconnectivity. To be more precise, we study the problem of disconnecting a maximally connected bipartite (di)graph by removing nontrivial subsets of vertices or edges. Within this framework, both an upper-bound on the diameter and Chartrand type conditions to guarantee optimum superconnectivities are obtained. Secondly, we show that if the order or size of a bipartite (di)graph is small enough then its vertex connectivity or edge-connectivity attain their maximum values. For example, a bipartite digraph is maximally edge-connected if \(\delta^+(x)+\delta^+(y)\geq \lceil\frac{n+1}{2}\rceil\) for all pair of vertices \(x, y\) such that \(d(x,y) \geq 4\). This result improves some conditions given by Dankelmann and Volkmann in [12] for the undirected case.