The vulnerability of the diameter of folded n-cubes

Abstract

Fault tolerance concern in the design of interconnection networks has arisen interest in the study of graphs such that the subgraphs obtained by deleting some vertices or edges have a moderate increment of the diameter. Besides the general problem, several particular families of graphs are worthy of consideration. Both the odd graphs and the n-cubes have been studied in this context. In this paper we deal with folded n-cubes, a much interesting family because: (i) like the n-cubes, their order is a power of 2, (ii) their diameter is half the diameter of the n-cube of the same order, while their degree only increases by one, and (iii) as we show, in a folded n-cube of degree Δ, the deletion of less than ⌊ 1 2 Δ⌋ − 1 vertices or edges does not increase the diameter of the graph, and the deletion of up to Δ − 1 vertices or edges increases it by at most one. This last property means that interconnection networks modelled by folded n-cubes are extremely robust.