Advertisement

A simple linear time algorithm for triangulating three-colored graphs

  • Hans Bodlaender
  • Ton Kloks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)

Abstract

In this paper we consider the problem of determining whether a given colored graph can be triangulated, such that no edges between vertices of the same color are added. This problem originated from the Perfect Phylogeny problem from molecular biology, and is strongly related with the problem of recognizing partial k-trees. In this paper we give a simple linear time algorithm that solves the problem when there are three colors. We do this by first giving a complete structural characterization of the class of partial 2-trees. We also give an algorithm that solves the problem for partial 2-trees.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H.L. Bodlaender, Classes of graphs with bounded tree-width, Tech. Rep. RUU-CS-86-22, Department of Computer Science, Utrecht University, Utrecht, 1986.Google Scholar
  2. [2]
    H.L. Bodlaender and T. Kloks, Better algorithms for the pathwidth and treewidth of graphs, Proceedings of the 18th International colloquium on Automata, Languages and Programming, 544–555, Springer Verlag, Lecture Notes in Computer Science, vol. 510 (1991).Google Scholar
  3. [3]
    G. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961).Google Scholar
  4. [4]
    M.R. Fellows and T. Warnow, Personal communication, 1991.Google Scholar
  5. [5]
    D. Fulkerson and O. Gross, Incidence matrices and interval graphs, Pacific J. Math. 15, 835–855 (1965).Google Scholar
  6. [6]
    M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.Google Scholar
  7. [7]
    S. Kannan and T. Warnow, Inferring Evolutionary History from DNA Sequences, in: Proceedings of the 31th Annual Symposium on Foundations of Computer Science, pp. 362–371, 1990.Google Scholar
  8. [8]
    S. Kannan and T. Warnow, Triangulating three-colored graphs, in: Proceedings of the 1st Ann. ACM-SIAM Symposium on Discrete Algorithms, pp. 337–343, 1990.Google Scholar
  9. [9]
    T. Kloks, Enumeration of biconnected partial 2-trees, to appear.Google Scholar
  10. [10]
    J. Matoušek and R. Thomas, Algorithms Finding tree-decompositions of graphs, Journal of Algorithms 12, 1–22 (1991).Google Scholar
  11. [11]
    C.F. McMorris, T. Warnow, and T. Wimer, Triangulating colored graphs, submitted to Inform. Proc. Letters.Google Scholar
  12. [12]
    J.A. Wald and C.J. Colbourn, Steiner trees, partial 2-trees and minimum IFI networks, Networks 13 (1983), 159–167.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Hans Bodlaender
  • Ton Kloks
    • 1
  1. 1.Department of Computer ScienceUtrecht UniversityTB UtrechtThe Netherlands

Personalised recommendations