Graphs and Combinatorics

, Volume 27, Issue 1, pp 61–72 | Cite as

Boxicity of Leaf Powers

  • L. Sunil Chandran
  • Mathew C. Francis
  • Rogers Mathew
Original Paper

Abstract

The boxicity of a graph G, denoted as boxi(G), is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are used in the construction of phylogenetic trees in evolutionary biology and have been studied in many recent papers. We show that for a k-leaf power G, boxi(G) ≤ k−1. We also show the tightness of this bound by constructing a k-leaf power with boxicity equal to k−1. This result implies that there exist strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.

Keywords

Boxicity Leaf powers Tree powers Strongly chordal graphs Interval graphs 

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References

  1. 1.
    Bhowmick, D., Sunil Chandran, L.: Boxicity of circular arc graphs. Preprint (2008)Google Scholar
  2. 2.
    Bohra A., Sunil Chandran L., Krishnam Raju J.: Boxicity of series parallel graphs. Discret. Math. 306(18), 2219–2221 (2006)MATHCrossRefGoogle Scholar
  3. 3.
    Brandstädt A., Bang Le V.: Structure and linear time recognition of 3-leaf powers. Inf. Process. Lett. 98, 133–138 (2006)MATHCrossRefGoogle Scholar
  4. 4.
    Brandstädt A., Bang Le V., Sritharan R.: Structure and linear-time recognition of 4-leaf powers. ACM Trans. Algorithm. 5(1), 1–22 (2008)CrossRefGoogle Scholar
  5. 5.
    Brandstädt, A., Wagner, P.: On (k, l)-leaf powers. In: Kucera, L., Kucera, A. (eds.), Mathematical Foundations of Computer Science 2007 (2007)Google Scholar
  6. 6.
    Chandran L.S., Francis M.C., Sivadasan N.: Boxicity and maximum degree. J. Comb. Theory Ser. B 98(2), 443–445 (2008)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chandran, L.S., Francis, M.C., Sivadasan, N.: Geometric representation of graphs in low dimension using axis parallel boxes. Algorithmica 56(2), 2010Google Scholar
  8. 8.
    Chandran L.S., Francis M.C., Suresh S.: Boxicity of Halin graphs. Discret. Math. 309(10), 3233–3237 (2009)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chandran L.S., Sivadasan N.: Boxicity and treewidth. J. Comb. Theory Ser. B 97(5), 733–744 (2007)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cozzens M.B., Roberts F.S.: Computing the boxicity of a graph by covering its complement by cointerval graphs. Discret. Appl. Math. 6, 217–228 (1983)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dahlhaus E., Duchet P.: On strongly chordal graphs. Ars Comb. 24B, 23–30 (1987)MathSciNetGoogle Scholar
  12. 12.
    Dahlhaus E., Manuel P.D., Miller M.: A characterization of strongly chordal graphs. Discret. Math. 187(1–3), 269–271 (1998)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Dom M., Guo J., Hüffner F., Niedermeier R.: Closest 4-leaf power is fixed-parameter tractable. Discret. Appl. Math. 156(18), 3345–3361 (2008)MATHCrossRefGoogle Scholar
  14. 14.
    Farber M.: Characterizations of strongly chordal graphs. Discret. Math. 43(2–3), 173–189 (1983)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kennedy W., Lin G., Yan G.: Strictly chordal graphs are leaf powers. J. Discret. Algorithms 4, 511–525 (2006)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lekkerkerker C.G., Boland J.C.: Representation of a finite graph by a set of intervals on the real line. Fundam. Math. 51, 45–64 (1962)MATHMathSciNetGoogle Scholar
  17. 17.
    Lubiw, A.: Γ-free matrices. Master’s thesis, Princeton University (1982)Google Scholar
  18. 18.
    McKee T.A.: Strong clique trees, neighborhood trees, and strongly chordal graphs. J. Graph Theory 33(3), 151–160 (2000)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    McKee T.A.: Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs. Discret. Math. 260(1–3), 231–238 (2003)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Nishimura N., Ragde P., Thilikos D.M.: On graph powers for leaf-labeled trees. J. Algorithms 42, 69–108 (2002)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Raychaudhuri A.: On powers of strongly chordal and circular graphs. Ars Comb. 34, 147–160 (1992)MATHMathSciNetGoogle Scholar
  22. 22.
    Roberts, F.S.: On the boxicity and cubicity of a graph. In: Recent Progresses in Combinatorics, pp. 301–310. Academic Press, New York (1969)Google Scholar
  23. 23.
    Scheinerman, E.R.: Intersection Classes and Multiple Intersection Parameters. PhD thesis, Princeton University (1984)Google Scholar
  24. 24.
    Thomassen C.: Interval representations of planar graphs. J. Comb. Theory Ser. B 40, 9–20 (1986)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • L. Sunil Chandran
    • 1
  • Mathew C. Francis
    • 1
  • Rogers Mathew
    • 1
  1. 1.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia

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