Many-to-Many Disjoint Path Covers in the Presence of Faulty Elements

  • Authors:

  • Jung-Heum Park;Hee-Chul Kim;Hyeong-Seok Lim

  • Affiliations:

  • The Catholic University of Korea, Korea;Hankuk University of Foreign Studies, Yongin-si;Chonnam National University, Gwangju

  • Venue:
  • IEEE Transactions on Computers
  • Year:
  • 2009

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Abstract

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k sources and k sinks in which each vertex of G is covered by a path. It is called a paired many-to-many disjoint path cover when each source should be joined to a specific sink, and it is called an unpaired many-to-many disjoint path cover when each source can be joined to an arbitrary sink. In this paper, we discuss about paired and unpaired many-to-many disjoint path covers including their relationships, application to strong Hamiltonicity, and necessary conditions. And then, we give a construction scheme for paired many-to-many disjoint path covers in the graph H_{0} \oplus H_{1} obtained from connecting two graphs H_{0} and H_{1} with |V(H_{0})| = |V(H_{1})| by |V(H_{0})| pairwise nonadjacent edges joining vertices in H_{0} and vertices in H_{1}, where H_{0} = G_{0} \oplus G_{1} and H_{1} = G_{2} \oplus G_{3} for some graphs G_{j}. Using the construction, we show that every m-dimensional restricted HL-graph and recursive circulant G(2^{m}, 4) with f or less faulty elements have a paired k-DPC for any f and k \geq 2 with f + 2k \leq m.