The fault tolerance of [formula omitted]-bubble-sort networks

Abstract

The h-good neighbor connectivity κh(G) is an important parameter to evaluate the fault tolerance of an interconnection network G. So far, almost all known results of κh(G) are about small h′s except the hypercubes, the star graphs, the k-ary n-cubes and hierarchical hypercube network HHCn et al. In this paper, we focus on κh(Bn,k) for the (n,k)-bubble-sort network Bn,k, which is a generalization of the bubble-sort graph Bn as it is a special case of (n,k)-bubble-sort networks for k=n−1. We show that κh(Bn,k)=n+h(k−2)−1 for 2≤k≤⌈n+32⌉ and 0≤h≤⌊n−32⌋, which implies that at least n+h(k−2)−1 vertices of Bn,k have to be deleted to get a disconnected graph without vertices of degree less than h.