On the restricted connectivity and superconnectivity in graphs with given girth

AbstractThe restricted connectivity κ′(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 κ1(G) is defined similarly, this time considering only vertices u in G-X, hence κ1(G)⩽κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u) standing for the degree of a vertex u. In this paper, several sufficient conditions yielding κ1(G)⩾ξ(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 κ1(G)=κ′(G)=ξ(G) under some additional constraints