The existence and upper bound for two types of restricted connectivity

In this paper, we study two types of restricted connectivity: κk(G)κk(G) is the cardinality of a minimum vertex cut SS such that every component of G−SG−S has at least kk vertices; κk′(G) is the cardinality of a minimum vertex cut SS such that there are at least two components in G−SG−S of order at least kk. In this paper, we give some sufficient conditions for the existence and upper bound of κk(G)κk(G) and/or κk′(G), and study some properties of these two parameters.