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

Abstract

In a graph G, k disjoint paths joining a single source and k distinct sinks that cover all the vertices in the graph are called a one-to-many k -disjoint path cover of G. We consider a k-disjoint path cover in a graph with faulty vertices and/or edges 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 a sufficient condition for such a graph to have a k-disjoint path cover and give the construction scheme. 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 k-disjoint path covers for any f ≥ 0 and k ≥ 2 such that f+k ≤ m–1.

This work was supported by grant No. R05-2003-000-11506-0 from the Basic Research Program of the Korea Science & Engineering Foundation.