Structure Connectivity and Substructure Connectivity of Alternating Group Graphs

The alternating group graph, denoted by AGn, is one of the popular interconnection networks. In this paper, we consider two network connectivities, H-structure-connectivity and H-substructure-connectivity, which are new measures for a network’s reliability and fault-tolerability. We say that a set F of connected subgraphs of G is a subgraph-cut of G if G−V (F) is a disconnected or trivial graph. Let H be a connected subgraph of G. Then F is an H-structure-cut, if F is a subgraph-cut, and every element in F is isomorphic to H. And F is an H-substructure-cut if F is a subgraph-cut, such that every element in F is isomorphic to 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 minimum cardinality of all H-structure-cuts(resp. H-substructure-cuts) of G. In this paper, we will establish both κ(AGn;H) and κ(AGn;H) for the alternating group graph AGn and H ∈{K1,K1,1,K1,2}.