The vulnerability of the diameter of the enhanced hypercubes

Abstract

Concern about fault tolerance in the design of interconnection networks has raised interest in the study of graphs such that deleting some vertices increases the diameter only moderately. For an interconnection network G, the (ω−1)-fault diameterDω(G) is the maximum diameter of a subgraph obtained by deleting fewer than ω vertices of G, and the ω-wide diameterdω(G) is the least ℓ such that any two vertices are joined by ω internally-disjoint paths of length at most ℓ. The enhanced hypercube Qn,k is a variant of the well-known n-dimensional hypercube Qn in which an edge is added from each vertex xn,…,x1 to the vertex obtained by complementing xk,…,x1. Yang, Chang, Pai, and Chan gave an upper bound for dn+1(Qn,k) and Dn+1(Qn,k) and posed the problem of finding the wide diameter and fault diameter of Qn,k. By constructing internally disjoint paths between any two vertices in the enhanced hypercube, for n≥3 and 2≤k≤n we prove that Dω(Qn,k)=dω(Qn,k)=d(Qn,k) for 1≤ω<n−⌊k2⌋; Dω(Qn,k)=dω(Qn,k)=d(Qn,k)+1 for n−⌊k2⌋≤ω≤n+1, where d(Qn,k) is the diameter of Qn,k. These results mean that interconnection networks modeled by enhanced hypercubes are extremely robust.