Unpaired Many-to-Many Disjoint Path Covers in Nonbipartite Torus-Like Graphs With Faulty Elements

Abstract

One of the key problems in parallel processing is finding disjoint paths in the underlying graph of an interconnection network. The disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. Given disjoint source and sink sets, $S = \{ s_{1},\ldots,s_{k}\}$ and $T = \{t_{1},\ldots, t_{k}\}$ , in graph $G$ , an unpaired many-to-many $k$ -disjoint path cover joining $S$ and $T$ is a disjoint path cover $\{ P_{1}, \ldots, P_{k}\}$ , in which each path $P_{i}$ runs from source $s_{i}$ to some sink $t_{j}$ . In this paper, we reveal that a nonbipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties of the unpaired type, retains such a good property. As a result, an $m$ -dimensional nonbipartite torus, $m \geq 2$ , with at most $f$ vertex and/or edge faults has an unpaired many-to-many $k$ -disjoint path cover joining arbitrary disjoint sets $S$ and $T$ of size $k$ each, subject to $k \geq 1$ and $f+k \leq 2m-2$ . The bound of $2m-2$ on $f+k$ is nearly optimal.

Keywords

• Disjoint path
• path cover
• path partition
• torus
• toroidal grid
• interconnection network