On complexity of substructure connectivity and restricted connectivity of graphs

The connectivity of a graph is an important parameter to evaluate its reliability. k-restricted connectivity (resp. R^h-restricted connectivity) of a graph G is the minimum cardinality of a set S of vertices in G, if exists, whose deletion disconnects G and leaves each component of $G-S$ with more than k vertices (resp. $\delta (G-S)\ge h$). In contrast, structure (substructure) connectivity of G is defined as the minimum number of vertex-disjoint subgraphs whose deletion disconnects G. As generalizations of the concept of connectivity, structure (substructure) connectivity, restricted connectivity and R^h-restricted connectivity have been extensively studied from the combinatorial point of view. Very little is known about the computational complexity of these variants, except for the recently established NP-completeness of k-restricted edge-connectivity. In this paper, we prove that the problems of determining structure, substructure, restricted, and R^h-restricted connectivity are all NP-complete.