Disjoint path covers in recursive circulants G(2m,4) with faulty elements

  • Authors:

  • Sook-Yeon Kim;Jae-Ha Lee;Jung-Heum Park

  • Affiliations:

  • Department of Computer Engineering, Hankyong National University, Ansung 456-749, Republic of Korea;Computer Engineering Department, Konkuk University, Seoul 143-701, Republic of Korea;School of Computer Science and Information Engineering, The Catholic University of Korea, Bucheon 420-743, Republic of Korea

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2011

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Abstract

A k-disjoint path cover of a graph is defined as a set of k internally vertex-disjoint paths connecting given sources and sinks in such a way that every vertex of the graph is covered by a path in the set. In this paper, we analyze the k-disjoint path cover of recursive circulant G(2^m,4) under the condition that at most f faulty vertices and/or edges are removed. It is shown that when m=3, G(2^m,4) has a k-disjoint path cover (of one-to-one type) joining any pair of two distinct source and sink for arbitrary f and k=2 subject to f+k@?m. In addition, it is proven that when m=5, G(2^m,4) has a k-disjoint path cover (of unpaired many-to-many type) joining any two disjoint sets of k sources and k sinks for arbitrary f and k=2 satisfying f+k@?m-1, in which sources and sinks are freely matched. In particular, the mentioned bounds f+k@?m and f+k@?m-1 of the two cases are shown to be optimal.