Structure connectivity and substructure connectivity of star graphs

The connectivity is an important measurement for the fault-tolerance of networks. The structure connectivity and substructure connectivity are two generalizations of the classical connectivity. For a fixed graph H, a set F of subgraphs of G is called an H-structure cut (resp., H-substructure cut) of G, if G−∪F∈FV(F) is disconnected and every element of F is isomorphic to H (resp., a connected subgraph of H). The H-structure connectivity (resp., H-substructure connectivity) of G, denoted by κ(G;H) (resp., κs(G;H)), is the cardinality of a minimal H-structure cut (resp., H-substructure cut) of G. In this paper, we will establish both κ(Sn;H) and κs(Sn,H) for every H∈{K1,K1,1,K1,2,…,K1,n−2,P4,P5,C6}, where Sn is the n-dimensional star graph. These results will show that star networks are highly tolerant of structure faults.