On the Connectivity and Superconnectivity of Bipartite Digraphs and Graphs

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 δ(x)+ δ−(y) ≥ dn+1 2 e for all pair of vertices x, y such that d(x, y) ≥ 4. This result improves some of the above known conditions given by Dankelmann and Volkmann in [10] for the undirected case.