Many-to-Many Disjoint Path Covers in a Graph with Faulty Elements

Abstract

In a graph G, k vertex disjoint paths joining k distinct source-sink pairs that cover all the vertices in the graph are called a many-to-many k -disjoint path cover(k-DPC) of G. We consider an f-fault k-DPC problem that is concerned with finding many-to-many k-DPC in the presence of f or less faulty vertices and/or edges. We consider the graph obtained by merging two graphs H 0 and H 1, |V(H 0)| = |V(H 1)| = n, with n pairwise nonadjacent edges joining vertices in H 0 and vertices in H 1. We present sufficient conditions for such a graph to have an f-fault k-DPC and give the construction schemes. Applying our main result to interconnection graphs, we observe that when there are f or less faulty elements, all of recursive circulant G(2 m ,4), twisted cube TQ m , and crossed cube CQ m of degree m have f-fault k-DPC for any k ≥ 1 and f ≥ 0 such that f + 2k ≤ m–1.