Many-to-Many Disjoint Path Covers in Hypercube-Like Interconnection Networks with Faulty Elements

• Authors: Jung-Heum Park;Hee-Chul Kim;Hyeong-Seok Lim
• Venue: IEEE Transactions on Parallel and Distributed Systems
• Year: 2006

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G_0 \oplus G_1 obtained from connecting two graphs G_0 and G_1 with n vertices each by n pairwise nonadjacent edges joining vertices in G_0 and vertices in G_1. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G_0 \oplus G_1 connecting two lower dimensional networks G_0 and G_1. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G_0 \oplus G_1 and (G_0 \oplus G_1) \oplus (G_2 \oplus G_3), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G_i are satisfied, 0 \leq i \leq 3. We apply our main results to recursive circulant G(2^m,4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclass includes twisted cubes, crossed cubes, multiply twisted cubes, Möbius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k{\hbox{-}}{\rm{DPC}} joining any k distinct source-sink pairs for any k \geq 1 and f \geq 0 such that f+2k\leq m-1.