On the restricted connectivity and superconnectivity in graphs with given girth

The restricted connectivity kappa'(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex u has all its neighbors in X; the superconnectivity kappa(1) is defined similarly, this time considering only vertices u in G - X, hence kappa(1)(G) <= kappa'(G). The minimum edge-degree of G is xi(G) = min{d(u) + d(v) - 2 : uv is an element of E(G)}, d(u) standing for the degree of a vertex u. In this paper, several sufficient conditions yielding kappa(1)(G) >= xi(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17-31] and guaranteeing kappa(1)(G) = kappa'(G) = xi(G) under some additional constraints. (c) 2006 Elsevier B.V. All rights reserved.