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On Graphs That Are Not PCGs

  • Stephane Durocher
  • Debajyoti Mondal
  • Md. Saidur Rahman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7748)

Abstract

Let T be an edge weighted tree and let d min ,d max be two nonnegative real numbers. Then the pairwise compatibility graph (PCG) of T is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves in T is in the interval [d min ,d max ]. Similarly, a given graph G is a PCG if there exist suitable T,d min ,d max , such that G is a PCG of T. Yanhaona, Bayzid and Rahman proved that there exists a graph with 15 vertices that is not a PCG. On the other hand, Calamoneri, Frascaria and Sinaimeri proved that every graph with at most seven vertices is a PCG. In this paper we construct a graph of eight vertices that is not a PCG, which strengthens the result of Yanhaona, Bayzid and Rahman, and implies optimality of the result of Calamoneri, Frascaria and Sinaimeri. We then construct a planar graph with sixteen vertices that is not a PCG. Finally, we prove a variant of the PCG recognition problem to be NP-complete.

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References

  1. 1.
    Brandstädt, A., Hundt, C., Mancini, F., Wagner, P.: Rooted directed path graphs are leaf powers. Discrete Mathematics 310(4), 897–910 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Brandstädt, A., Le, V.B., Rautenbach, D.: Exact leaf powers. Theoretical Computer Science 411(31-33), 2968–2977 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brandstädt, A., Wagner, P.: Characterising (k, l)-leaf powers. Discrete Applied Mathematics 158(2), 110–122 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Calamoneri, T., Frascaria, D., Sinaimeri, B.: All graphs with at most seven vertices are pairwise compatibility graphs. The Computer Journal (to appear, 2012), http://arxiv.org/abs/1202.4631
  5. 5.
    Calamoneri, T., Petreschi, R., Sinaimeri, B.: On Relaxing the Constraints in Pairwise Compatibility Graphs. In: Rahman, M. S., Nakano, S.-I. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 124–135. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Fellows, M.R., Meister, D., Rosamond, F.A., Sritharan, R., Telle, J.A.: Leaf Powers and Their Properties: Using the Trees. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 402–413. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Kearney, P.E., Munro, J.I., Phillips, D.: Efficient Generation of Uniform Samples from Phylogenetic Trees. In: Benson, G., Page, R.D.M. (eds.) WABI 2003. LNCS (LNBI), vol. 2812, pp. 177–189. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Kennedy, W.S., Lin, G., Yan, G.: Strictly chordal graphs are leaf powers. Journal of Discrete Algorithms 4(4), 511–525 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Nishimura, N., Ragde, P., Thilikos, D.M.: On graph powers for leaf-labeled trees. Journal of Algorithms 42(1), 69–108 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Salma, S.A., Rahman, M.S.: Triangle-Free Outerplanar 3-Graphs Are Pairwise Compatibility Graphs. In: Rahman, M. S., Nakano, S.-I. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 112–123. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of Symposium on Theory of Computing, STOC 1978, pp. 216–226 (1978)Google Scholar
  12. 12.
    Yanhaona, M.N., Bayzid, M.S., Rahman, M.S.: Discovering pairwise compatibility graphs. Discrete Mathematics, Algorithms and Applications 2(4), 607–623 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Yanhaona, M.N., Hossain, K.S.M.T., Rahman, M.S.: Pairwise compatibility graphs. Journal of Applied Mathematics and Computing 30(1-2), 479–503 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Debajyoti Mondal
    • 1
  • Md. Saidur Rahman
    • 2
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada
  2. 2.Department of Computer Science and Engineering, Graph Drawing & Information Visualization LaboratoryBangladesh University of Engineering and TechnologyBangladesh

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