Super fault-tolerance assessment of locally twisted cubes based on the structure connectivity

Abstract

The connectivity of a connected graph G is the minimum cardinality over all vertex-cuts, which can be determined using Menger's theorem (1927). Then G is super connected if its minimum vertex-cut is always composed of a vertex's neighborhood. Two generalized extensions to this classic notion of connectivity include the T-structure connectivity κ(G;T) and the T-substructure connectivity κs(G;T), for which T is the given structure isomorphic to a connected subgraph of G. Let T⁎ denote the union of the set of all connected subgraphs of T and the set of the trivial graph. In this article, a connected graph G is called super T-connected if the minimum degree of G−F is zero for each minimum T-cut F of G; analogously, G is super T⁎-connected if the minimum degree of G−F is zero for each minimum T⁎-cut F of G. Considering the n-dimensional locally twisted cube LTQn with n≥3, we first establish both κ(LTQn;T) and κs(LTQn;T) and then determine whether LTQn is super T-connected and super T⁎-connected, where T∈{K1,1,K1,2,K1,3,C4}∪{Pk|4≤k≤n}.