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On Dilworth k Graphs and Their Pairwise Compatibility

  • Tiziana Calamoneri
  • Rossella Petreschi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)

Abstract

The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another one. In this paper we give a new characterization of Dilworth k graphs, for each value of k, exactly defining their structure. Moreover, we put these graphs in relation with pairwise compatibility graphs (PCGs), i.e. graphs on n nodes that can be generated from an edge-weighted tree T that has n leaves, each representing a different node of the graph; two nodes are adjacent in the graph if and only if the weighted distance in the corresponding T is between two given non-negative real numbers, m and M. When either m or M are not used to eliminate edges from G, the two subclasses leaf power and minimum leaf power graphs (LPGs and mLPGs, respectively) are defined. Here we prove that graphs that are either LPGs or mLPGs of trees obtained connecting the centers of k stars with a path are Dilworth k graphs. We show that the opposite is true when k = 1,2, but not when k ≥ 3. Finally, we show that the relations we proved between Dilworth k graphs and chains of k stars hold only for LPGs and mLPGs, but not for PCGs.

Keywords

Graphs with Dilworth number k leaf power graphs minimum leaf power graphs pairwise compatibility graphs 

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References

  1. 1.
    Belmonte, R., Vatshelle, M.: Graph Classes with Structured Neighborhoods and Algorithmic Applications. Theoretical Computer Science (2013), doi:10.1016/j.tcs.2013.01.011Google Scholar
  2. 2.
    Benzaken, C., Hammer, P.L., de Werra, D.: Threshold characterization of graphs with Dilworth number 2. Journal of Graph Theory 9, 245–267 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brandstädt, A.: On Leaf Powers. Technical report, University of Rostock (2010)Google Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  5. 5.
    Brandstädt, A., Hundt, C.: Ptolemaic Graphs and Interval Graphs Are Leaf Powers. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 479–491. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Calamoneri, T., Frascaria, D., Sinaimeri, B.: All graphs with at most seven vertices are Pairwise Compatibility Graphs. The Computer Journal 56(7), 882–886 (2013)CrossRefGoogle Scholar
  7. 7.
    Calamoneri, T., Montefusco, E., Petreschi, R., Sinaimeri, B.: Exploring Pairwise Compatibility Graphs. Theoretical Computer Science 468, 23–36 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Calamoneri, T., Petreschi, R.: Graphs with Dilworth Number Two are Pairwise Compatibility Graphs. Lagos 2013, Electronic Notes in Discrete Mathematics 44, 31–38 (2013)Google Scholar
  9. 9.
    Calamoneri, T., Petreschi, R., Sinaimeri, B.: On the Pairwise Compatibility Property of some Superclasses of Threshold Graphs. In: Special Issue of WALCOM 2012 on Discrete Mathematics, Algorithms and Applications (DMAA), vol. 5(2) (2013) (to appear)Google Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Chvatal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Annals of Discrete Math. 1, 145–162 (1977)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Durocher, S., Mondal, D., Rahman, M.S.: On Graphs that are not PCGs. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 310–321. Springer, Heidelberg (2013)Google Scholar
  13. 13.
    Felsner, S., Raghavan, V., Spinrad, J.: Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number. Order 20(4), 351–364 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Foldes, S., Hammer, P.L.: The Dilworth number of a graph. Annals of Discrete Math. 2, 211–219 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hammer, P., Peled, U.N., Sun, X.: Difference graphs. Discrete Applied Math. 28(1), 35–44 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hoang, C.T., Mahadev, N.V.R.: A note on perfect orders. Discrete Math. 74, 77–84 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kearney, P.E., Corneil, D.G.: Tree powers. J. Algorithms 29(1) (1998)Google Scholar
  18. 18.
    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
  19. 19.
    Lin, G., Kearney, P.E., Jiang, T.: Phylogenetic k-Root and Steiner k-Root. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 539–551. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Mahadev, N.V.R., Peled, U.N.: Threshold Graphs and Related Topics. Ann. Discrete Math., vol. 56. North-Holland, Amsterdam (1995)Google Scholar
  21. 21.
    Payan, C.: Perfectness and Dilworth number. Discrete Math. 44, 229–230 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Phillips, D.: Uniform sampling from phylogenetics trees. Masters Thesis, University of Waterloo (2002)Google Scholar
  23. 23.
    Yanhaona, M.N., Hossain, K.S.M.T., Rahman, M.S.: Pairwise compatibility graphs. J. Appl. Math. Comput. 30, 479–503 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Yanhaona, M.N., Hossain, K.S.M.T., Rahman, M.S.: Ladder graphs are pairwise compatibility graphs. In: AAAC 2011 (2011)Google Scholar
  25. 25.
    Yanhaona, M.N., Bayzid, M.S., Rahman, M.S.: Discovering Pairwise compatibility graphs. Discrete Mathematics, Algorithms and Applications 2(4), 607–623 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Tiziana Calamoneri
    • 1
  • Rossella Petreschi
    • 1
  1. 1.Department of Computer Science“Sapienza” University of RomeItaly

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