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)


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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  1. 1.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41, 153–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cornell, D.G., Perl, Y., Stewart, L.K.: A Linear recognition algorithm for cographs. SIAM Jl. on Computing 14, 926–934 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cornuejols, G., Hartvigsen, D., Pulleyblank, W.: Packing Subgraphs in a Graph. Operations Research Letters 1, 139–143 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dahlhaus, E., Karpinski, M.: Matching and Multidimensional Matching in Chordal and Strongly Chordal Graphs. Discerete Applied Math. (84), 79–91 (1998)Google Scholar
  5. 5.
    Edmonds, J.: Paths, trees and flowers. Canadian J. Math. 17, 449–469 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the theory of NP-completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  7. 7.
    Golumbic, M.C.: Algorithmic graph theory and Perfect graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  8. 8.
    Harary, F.: Graph Theory. Addison- Wesley, Reading (1969)zbMATHGoogle Scholar
  9. 9.
    Hell, P., Kirkpatrick, D.G.: On generalized matching problems. Info. Proc. Letters 12, 33–35 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: A unified approach to approximation schemes for NP- and PSPACE-hard problems for geometric graphs. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 424–435. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  11. 11.
    Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Mathematics 2, 68–72 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters 37, 27–35 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kirkpatrick, D.G., Hell, P.: On the complexity of general graph factor problems. SIAM JI. on Computing 12, 601–609 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Micali, S., Vazirani, V.V.: An O(\(\sqrt{|V|}{|E|}\)) algorithm for finding maximum matching in general graphs. In: Proc. 21st Annual Symposium on the foundation of Comp. Sci., pp. 17–27 (1980)Google Scholar

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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