The existence and upper bound for two types of restricted connectivity

In this paper, we study two types of restricted connectivity: kappa(k)(G) is the cardinality of a minimum vertex cut S such that every component of G-S has at least k vertices; kappa(k)'(G) is the cardinality, of a minimum vertex cut 5 such that there are at least two components in G - S of order at least k. In this paper, we give some sufficient conditions for the existence and upper bound of kappa(k)(G) and/or kappa(k)'(G), and study some properties of these two parameters. (C) 2009 Elsevier B.V. All rights reserved.