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Issue No.03 - March (2006 vol.17)

pp: 227-240

Jung-Heum Park , IEEE

Hyeong-Seok Lim , IEEE

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TPDS.2006.37

ABSTRACT

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

INDEX TERMS

Fault tolerance, network topology, graph theory, fault-Hamiltonicity, embedding, strong Hamiltonicity, recursive circulants, restricted HL-graphs.

CITATION

Jung-Heum Park, Hee-Chul Kim, Hyeong-Seok Lim, "Many-to-Many Disjoint Path Covers in Hypercube-Like Interconnection Networks with Faulty Elements",

*IEEE Transactions on Parallel & Distributed Systems*, vol.17, no. 3, pp. 227-240, March 2006, doi:10.1109/TPDS.2006.37