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Ptolemaic Graphs and Interval Graphs Are Leaf Powers

  • Andreas Brandstädt
  • Christian Hundt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

Abstract

Motivated by the problem of reconstructing evolutionary history, Nishimura, Radge and Thilikos introduced the notion of k-leaf powers as the class of graphs G = (V,E) which have a k-leaf root, i.e., a tree T with leaf set V where xy ∈ E if and only if the T-distance between x and y is at most k. It is known that leaf powers are strongly chordal (i.e., sun-free chordal) graphs. Despite extensive research, the problem of recognizing leaf powers, i.e., to decide for a given graph G whether it is a k-leaf power for some k, remains open. Much less is known on the complexity of finding the leaf rank of G, i.e., to determine the minimum number k such that G is a k-leaf power. A result by Bibelnieks and Dearing implies that not every strongly chordal graph is a leaf power. Recently, Kennedy, Lin and Yan have shown that dart- and gem-free chordal graphs are 4-leaf powers. We generalize their result and show that ptolemaic (i.e., gem-free chordal) graphs are leaf powers. Moreover, ptolemaic graphs have unbounded leaf rank. Furthermore, we show that interval graphs are leaf powers which implies that leaf powers have unbounded clique-width. Finally, we characterize unit interval graphs as those leaf powers having a caterpillar leaf root.

Keywords and Classification

Leaf powers leaf roots strongly chordal graphs ptolemaic graphs graph powers graph class inclusions (unit) interval graphs clique-width 

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References

  1. 1.
    Bandelt, H.J., Mulder, H.M.: Distance hereditary graphs. J. Combinatorial Theory (B) 41, 182–208 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bibelnieks, E., Dearing, P.M.: Neighborhood subtree tolerance graphs. Discrete Applied Math. 43, 13–26 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Le, V.B.: Structure and linear time recognition of 3-leaf powers. Information Processing Letters 98, 133–138 (2006)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Rautenbach, D.: Distance-hereditary 5-leaf powers (manuscript, 2006)Google Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. In: SIAM Monographs on Discrete Math. Appl., vol. 3, SIAM, Philadelphia (1999)Google Scholar
  6. 6.
    Brandstädt, A., Le, V.B., Sritharan, R.: Structure and linear time recognition of 4-leaf powers (manuscript, 2006)Google Scholar
  7. 7.
    Brandstädt, A., Wagner, P.: On (k,ℓ)-Leaf Powers. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 525–535. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Broin, M.W., Lowe, T.J.: A dynamic programming algorithm for covering problems with (greedy) totally balanced constraint matrices. SIAM J. Alg. Disc. Meth. 7, 348–357 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Chang, M.-S., Ko, T.: The 3-Steiner Root Problem. In: Extended abstract in: Proceedings 33rd International Workshop on Graph-Theoretic Concepts in Computer Science WG 2007. LNCS, vol. 4769, pp. 109–120. Springer, Heidelberg (2007)Google Scholar
  10. 10.
    Chen, Z.-Z., Jiang, T., Lin, G.: Computing phylogenetic roots with bounded degrees and errors. SIAM J. Computing 32, 864–879 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique width. Theory of Computing Systems 33, 125–150 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Dahlhaus, E., Duchet, P.: On strongly chordal graphs. Ars Combinatoria 24 B, 23–30 (1987)MathSciNetGoogle Scholar
  13. 13.
    Deogun, J.S., Gopalakrishnan, K.: Consecutive Retrieval Property - Revisited. Information Processing Letters 69, 15–20 (1999)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Niedermeier, R., Guo, J., Hüffner, F., Dom, M.: Error Compensation in Leaf Root Problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 389–401. Springer, Heidelberg (2004); Algorithmica 44(4), 363-381 (2006)Google Scholar
  15. 15.
    Fagin, R.: A painless introduction. In: Protasi, M., Ausiello, G. (eds.) CAAP 1983. LNCS, vol. 159, pp. 65–89. Springer, Heidelberg (1983)Google Scholar
  16. 16.
    Farber, M.: Characterizations of strongly chordal graphs. Discrete Math. 43, 173–189 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Golumbic, M., Rotics, U.: On the clique-width of some perfect graph classes. International J. Foundat. Computer Science 11(3), 423–443 (2000)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Gurski, F., Wanke, E.: The clique-width of tree powers and leaf-power graphs. Extended abstract In: Proceedings 33rd International Workshop on Graph-Theoretic Concepts in Computer Science WG 2007. LNCS, vol. 4769, pp. 76–85. Springer, Heidelberg (2007)Google Scholar
  19. 19.
    Howorka, E.: A characterization of distance-hereditary graphs. Quart. J. Math. Oxford 2(28), 417–420 (1977)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Howorka, E.: A characterization of ptolemaic graphs. J. Graph Theory 5, 323–331 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Kennedy, W., Lin, G., Yan, G.: Strictly chordal graphs are leaf powers. J. Discrete Algorithms 4, 511–525 (2006)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kloks, T.: Private communication (2007)Google Scholar
  23. 23.
    Lin, G.-H., 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
  24. 24.
    Lubiw, A.: Γ-free matrices, Master Thesis, Dept. of Combinatorics and Optimization, University of Waterloo, Canada (1982)Google Scholar
  25. 25.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. In: SIAM Monographs on Discrete Math. Appl., vol. 2, SIAM, Philadelphia (1999)Google Scholar
  26. 26.
    Nishimura, N., Ragde, P., Thilikos, D.M.: On graph powers for leaf labeled trees. J. Algorithms 42, 69–108 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Rautenbach, D.: Some remarks about leaf roots. Discrete Math. 306(13), 1456–1461 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Raychaudhuri, A.: On powers of strongly chordal and circular arc graphs. Ars Combinatoria 34, 147–160 (1992)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Roberts, F.S.: Representations of Indifference Relations, Ph.D. thesis, Standford University, Standford, CA (1968)Google Scholar
  30. 30.
    Todinca, I.: Coloring powers of graphs of bounded clique-width. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 370–382. Springer, Heidelberg (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andreas Brandstädt
    • 1
  • Christian Hundt
    • 1
  1. 1.Institut für Informatik, Universität Rostock, D-18051Germany

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