Extraconnectivity of graphs with large minimum degree and girth

The extraconnectivity ~,-(n) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to K(0) and 1<(1), respectively. This paper gives sufficient conditions, relating the diameter D, the girth g, and the minimum degree 6 of a graph, to assure maximum extraconnectivity. For instance, if D ~<; g n + 2(6 3), tbr n >~ 26 + 4 and ,q ~> n + 5, then the value of to(n) is (n + 1)6 2n, which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity 2(n), is also discussed.