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 boolean OR V-F is an element of F(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 kappa(G; H) (resp., kappa(s)(G; H)), is the cardinality of a minimal H-structure cut (resp., H-substructure cut) of G. In this paper, we will establish both kappa(S-n; H) and kappa(s)(S-n, H) for every H is an element of {K-1, K-1,K-1, K-1(,2), ..., K-1(,n-2,) P-4(,) P-5, C-6}, where S-n is the n-dimensional star graph. These results will show that star networks are highly tolerant of structure faults. (C) 2020 Elsevier B.V. All rights reserved.