Structure connectivity and substructure connectivity of wheel networks

Abstract

Consider a graph G and its connected subgraph T. The T-structure connectivity κ(G;T) of G is the cardinality of a minimum set of subgraphs in G, whose removal disconnects G and each element in the set is isomorphic to T. The T-substructure connectivity κs(G;T) of G is the cardinality of a minimum set of subgraphs in G, whose removal disconnects G and each element in the set is isomorphic to a connected subgraph of T. In G, the standard connectivity κ(G) is regarded as a simplification of both κ(G;T) and κs(G;T). The wheel network, denoted by CWn, is an attractive interconnected network prototype for multiple CPU systems. In this paper, we determine κ(CWn;P2k+1) (resp. κs(CWn;P2k+1)) for n≥5 and k+1≤2n−4, κ(CWn;P2k) (resp. κs(CWn;P2k)) for n≥6 and k≤2n−4 and a lower bound of κ(CWn;C2k) (resp. κs(CWn;C2k)) for n≥6 and k≤2n−4.