The diameter vulnerability of two-dimensional optimal circulant networks

Abstract

This paper studies the effect of changing the diameter of the network in circulant networks of dimension two with unreliable elements (nodes, links). The well-known (Δ,D,D′, s)-problem is to find (Δ, D)-graphs with maximum degree Δ and diameter D such that the subgraphs obtained from the original graph by deleting any set of up to s vertices (edges) have diameter at most D′. For a family of optimal circulants of degree four we found the ranges of the orders of the graphs that preserve the diameter of the graph for one (two) vertex or edge failures. It is proved that in the investigated circulant networks in case of failure of one or two edges (vertices), the diameter can increase by no more than one, and in case of failure of three elements by no more than two (edge failures) or three (vertex failures). It is shown that two-dimensional optimal circulants, in comparison with two-dimensional tori, have a better diameter in case of element failures.