A new characterization of P4-connected graphs

  • Luitpold Babel
  • Stephan Olariu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1197)


A graph is said to be P4-connected if for every partition of its vertices into two nonempty disjoint sets, some P4 in the graph contains vertices from both sets in the partition. A P4-chain is a sequence of vertices such that every four consecutive ones induce a P4. The main result of this work states that a graph is P4-connected if and only if each pair of vertices is connected by a P4chain. Our proof relies, in part, on a linear-time algorithm that, given two distinct vertices, exhibits a P4-chain connecting them. In addition to shedding new light on the structure of P4-connected graphs, our result extends a previously known theorem about the P4-structure of unbreakable graphs.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Babel, S. Olariu, On the isomorphism of graphs with few P 4s, in: M. Nagl, ed., Graph-Theoretic Concepts in Computer Science, 21th International Workshop, WG'95, Lecture Notes in Computer Science 1017, 24–36, Springer-Verlag, Berlin 1995.Google Scholar
  2. 2.
    J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North-Holland, Amsterdam, 1976.Google Scholar
  3. 3.
    V. Chvátal, A semi-strong perfect graph conjecture, Annals of Discrete Mathematics 21 (1984) 279–280.Google Scholar
  4. 4.
    V. Chvátal, Star-cutsets and perfect graphs. Journal of Combinatorial Theory (B) 39 (1985) 189–199.Google Scholar
  5. 5.
    V. Chvátal, On the P 4-structure of perfect graphs III. Partner decompositions, Journal of Combinatorial Theory (B) 43 (1987) 349–353.Google Scholar
  6. 6.
    A. Cournier, M. Habib, A new linear time algorithm for modular decomposition, Trees in Algebra and Programming, Lecture Notes in Computer Science 787, 68–84, Springer-Verlag 1994.Google Scholar
  7. 7.
    M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.Google Scholar
  8. 8.
    B. Jamison, S. Olariu, P 4-reducible graphs, a class of uniquely tree representable graphs, Studies in Applied Mathematics 81 (1989) 79–87.Google Scholar
  9. 9.
    B. Jamison, S. Olariu, On a unique tree representation for P 4-extendible graphs, Discrete Applied Mathematics 34 (1991) 151–164.Google Scholar
  10. 10.
    B. Jamison, S. Olariu, A unique tree representation for P 4-sparse graphs, Discrete Applied Mathematics 35 (1992) 115–129.Google Scholar
  11. 11.
    B. Jamison, S. Olariu, p-components and the homogeneous decomposition of graphs, SIAM Journal on Discrete Mathematics 8 (1995) 448–463.Google Scholar
  12. 12.
    R. McConnell, J. Spinrad, Linear-time modular decomposition and efficient transitive orientation of comparability graphs, Fifth Annual ACM-SIAM Symposium of Discrete Algorithms 536–545, 1994.Google Scholar
  13. 13.
    R. H. Möhring, Algorithmic aspects of comparability graphs and interval graphs, in: I. Rival, ed., Graphs and Orders, Dordrecht, Holland, 1985.Google Scholar
  14. 14.
    S. Olariu, On the structure of unbreakable graphs, Journal of Graph Theory 15 (1991) 349–373.Google Scholar
  15. 15.
    B.A. Reed, A semi-strong perfect graph theorem, Journal of Combinatorial Theory (B) 43 (1987) 223–240.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Luitpold Babel
    • 1
  • Stephan Olariu
    • 2
  1. 1.Technische Universität MünchenMünchenGermany
  2. 2.Old Dominion UniversityNorfolkUSA

Personalised recommendations