Connectedness of digraphs and graphs under constraints on the conditional diameter

Given a digraph G with minimum degree δ and an integer 0 ≤ ν ≤ δ, consider every pair of vertex subsets V1 and V2 such that both the minimum out-degree of the induced subdigraph G[V1] and the minimum in-degree of G[V2] are at least ν. The conditional diameter Dν of G is defined as the maximum of the distances d(V1, V2) between any two such vertex subsets. Clearly, D0 is the standard diameter and D0 ≥ D1 ≥ · · · ≥ Dδ holds. In this article, we guarantee appropriate lower bounds for the connectivities and superconnectivities of a digraph G when Dν ≤ h( π ), h( π ) being a function of the parameter π—which is related to the shortest paths in G. As a corollary of these results, we give some constraints of the kind Dν ≤ h( π ), which assure that the digraph is maximally connected, maximally edge-connected, superconnected, or edgesuperconnected, extending other previous results of the same kind. Similar statements can be obtained for a graph as a direct consequence of those for its associated symmetric digraph. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 80–87 2005