Characterizations of Minimum Structure- and Substructure-Cuts of Hypercubes

Abstract

Abstract For a graph G and a connected subgraph H of G, denote by ℋ(G;H) the set of subgraphs of G which are isomorphic to H and denote by ℋS(G;H) the union of sets of subgraphs of T, where T ranges over all the elements in ℋ(G;H). An H-structure-cut (respectively, H-substructure-cut) of G is a subset of ℋ(G;H) (respectively, ℋS(G;H)), if any, whose removal disconnects G. The H-structure connectivity (respectively, H-substructure connectivity) is the cardinality of a minimum H-structure-cut (respectively, H-substructure-cut) of G. The hypercube is one of the most attractive interconnection networks for large-scale multiprocessor computer systems. In this paper, we will characterize the minimum H-structure-cuts and the minimum H-substructure-cuts of hypercubes for H∈{K1,1,K1,2,K1,3,C4}.