Two Results on Intersection Graphs of Polygons

  • Jan Kratochvíl
  • Martin Pergel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)

Abstract

Intersection graphs of convex polygons inscribed to a circle, so called polygon-circle graphs, generalize several well studied classes of graphs, e.g., interval graphs, circle graphs, circular-arc graphs and chordal graphs. We consider the question how complicated need to be the polygons in a polygon-circle representation of a graph.

Let cmp (n) denote the minimum k such that every polygon-circle graph on n vertices is the intersection graph of k-gons inscribed to the circle. We prove that cmp(n) = n − log 2 n + o(log 2 n) by showing that for every positive constant c < 1,cmp(n) ≤ n − clogn for every sufficiently large n, and by providing an explicit construction of polygon-circle graphs on n vertices which are not representable by polygons with less than n − logn − 2 loglogn corners. We also show that recognizing intersection graphs of k-gons inscribed in a circle is an NP-complete problem for every fixed k ≥ 3.

References

  1. 1.
    Bouchet, A.: Circle graph obstructions. J. Comb. Theory, Ser. B 60(1), 107–144 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Eschen, E.M., Spinrad, J.P.: An O(n 2) algorithm for circular-arc graph recognition. In: Proceedings 4th ACM-SIAM Symposium on Discrete Algorithms, pp. 128–137 (1993)Google Scholar
  3. 3.
    de Frayssiex, H.: A characterization of circle graphs. Eur. J. Comb. 5, 223–238 (1984)Google Scholar
  4. 4.
    Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gyarfás, A.: Problems from the world surrounding perfect graphs. Zastosow. Mat. 19(3/4), 413–441 (1987)MATHGoogle Scholar
  6. 6.
    Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10, 713–717 (1981)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Koebe, M.: On a New Class of Intersection Graphs. In: Proceedings of the Fourth Czechoslovak Symposium on Combinatorics Prachatice, pp. 141–143 (1990)Google Scholar
  8. 8.
    Kratochvíl, J.: String graphs II. Recognizing string graphs is NP-hard. J. Combin. Theory Ser. B 52, 67–78 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kratochvíl, J., Kostochka, A.: Covering and coloring polygon-circle graphs. Discrete Math. 163, 299–305 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kratochvíl, J., Lubiw, A., Nešetřil, J.: Noncrossing subgraphs of topological layouts. SIAM J. Discrete Math. 4, 223–244 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kratochvíl, J., Matoušek, J.: String graphs requiring exponential representations. J. Combin. Theory Ser. B 53, 1–4 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pach, J., Tóth, G.: Recognizing string graphs is decidable. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 247–260. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Spinrad, J.P.: Recognition of Circle Graphs. Journal of Algorithms 16(1), 264–282 (1994)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
  15. 15.
    Schaefer, M., Stefankovič, D.: Decidability of string graphs. In: STOC 2001, Proceedings 33rd Annual ACM Symposium on Theory of Computing, Greece, pp. 241–246 (2001)Google Scholar
  16. 16.
    Schaefer, M., Sedgwick, E., Stefankovič, D.: Recognizing string graphs in NP. In: STOC 2002, Proceedings 34th Annual ACM Symposium on Theory of Computing (2002)Google Scholar
  17. 17.
    Tucker, A.C.: An efficient test for circular-arc graphs. SIAM J. Comput. 9, 1–24 (1980)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jan Kratochvíl
    • 1
  • Martin Pergel
    • 1
  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer ScienceCharles UniversityPraha 1Czech Republic

Personalised recommendations