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The Vertex-Disjoint Triangles Problem

  • Venkatesan Guruswami
  • C. Pandu Rangan
  • M. S. Chang
  • G. J. Chang
  • C. K. Wong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)

Abstract

The vertex-disjoint triangles (VDT) problem asks for a set of maximum number of pairwise vertex-disjoint triangles in a given graph G. The triangle cover problem asks for the existence of a perfect triangle packing in a graph G. It is known that the triangle cover problem is NP-complete on general graphs with clique number 3 [6]. The VDT problem is MAX SNP-hard on graphs with maximum degree four, while it can be approximated within 3/2+ε, for any ε > 0, in polynomial time [11].

We prove that the VDT problem is NP-complete even when the input graphs are chordal, planar, line or total graphs. We present an \(O(m \sqrt{n})\) algorithm for the VDT problem on split graphs and an O(n3) algorithm for the VDT problem on cographs. A linear algorithm for the triangle cover problem on strongly chordal graphs is also presented. Finally, the notion of packing-hardness, which may be crucial to the understanding of the difficulty of generalized matching problems, is defined.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • C. Pandu Rangan
    • 1
  • M. S. Chang
    • 2
  • G. J. Chang
    • 3
  • C. K. Wong
    • 4
  1. 1.Dept. of Computer Science & EnggIndian Institute of TechnologyMadrasIndia
  2. 2.Dept. of Computer Science and Information Engg.National Chung Cheng UniversityTaiwanRepublic of China
  3. 3.Dept. of Computer ScienceNational Chio-Tung UniversityTaiwanRepublic of China
  4. 4.Dept. of Computer Science and EnggChinese University of Hong KongHong Kong

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