Maximum Number of Minimal Feedback Vertex Sets in Chordal Graphs and Cographs

  • Jean-François Couturier
  • Pinar Heggernes
  • Pim van ’t Hof
  • Yngve Villanger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

A feedback vertex set in a graph is a set of vertices whose removal leaves the remaining graph acyclic. Given the vast number of published results concerning feedback vertex sets, it is surprising that the related combinatorics appears to be so poorly understood. The maximum number of minimal feedback vertex sets in a graph on n vertices is known to be at most 1.864 n . However, no examples of graphs having 1.593 n or more minimal feedback vertex sets are known, which leaves a considerable gap between these upper and lower bounds on general graphs. In this paper, we close the gap completely for chordal graphs and cographs, two famous perfect graph classes that are not related to each other. We prove that for both of these graph classes, the maximum number of minimal feedback vertex sets is \(10^{\frac{n}{5}} \approx 1.585^n\), and there is a matching lower bound.

Keywords

Disjoint Union Maximal Clique Chordal Graph Graph Class Perfect Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blair, J.R.S., Peyton, B.W.: An Introduction to Chordal Graphs and Clique Trees. In: Graph Theory and Sparse Matrix Computations. IMA Vol. in Math. Appl., vol. 56, pp. 1–27. SpringerGoogle Scholar
  2. 2.
    Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  3. 3.
    Bui-Xuan, B.M., Telle, J.A., Vatshelle, M.: Feedback Vertex Set on Graphs of low Cliquewidth. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 113–124. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Buneman, P.: A characterization of rigid circuit graphs. Disc. Math. 9, 205–212 (1974)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Corneil, D.G., Perl, Y., Stewart, L.: A linear recognition algorithm for cographs. SIAM J. Computing 14, 926–934 (1985)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Corneil, D.G., Fonlupt, J.: The complexity of generalized clique covering. Disc. Appl. Math. 22, 109–118 (1988/1989)Google Scholar
  7. 7.
    Couturier, J.-F., Heggernes, P., van ’t Hof, P., Kratsch, D.: Minimal Dominating Sets in Graph Classes: Combinatorial Bounds and Enumeration. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 202–213. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Eppstein, D.: Small maximal independent sets and faster exact graph coloring. J. Graph Algor. Appl. 7(2), 131–140 (2003)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1) (2008)Google Scholar
  11. 11.
    Fomin, F.V., Heggernes, P., Kratsch, D., Papadopoulos, C., Villanger, Y.: Enumerating Minimal Subset Feedback Vertex Sets. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 399–410. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Texts in Theoretical Computer Science (2010)Google Scholar
  13. 13.
    Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: Proceedings STACS 2010, pp. 383–394 (2010)Google Scholar
  14. 14.
    Gaspers, S., Mnich, M.: On Feedback Vertex Sets in Tournaments. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 267–277. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Disc. Math. 57 (2004)Google Scholar
  16. 16.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: Enumeration of Minimal Dominating Sets and Variants. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 298–309. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Lawler, E.L.: A note on the complexity of the chromatic number problem. Inform. Proc. Lett. 5, 66–67 (1976)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    McConnell, R.: Linear-time recognition of circular-arc graphs. Algorithmica 37, 93–147 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Okamoto, Y., Uehara, R., Uno, T.: Counting the Number of Matchings in Chordal and Chordal Bipartite Graph Classes. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 296–307. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. 21.
    Okamoto, Y., Uno, T., Uehara, R.: Counting the number of independent sets in chordal graphs. J. Disc. Alg. 6, 229–242 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schwikowski, B., Speckenmeyer, E.: On enumerating all minimal solutions of feedback problems. Disc. Appl. Math. 117, 253–265 (2002)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Spinrad, J.P.: Efficient graph representations. AMS, Fields Institute Monograph Series 19 (2003)Google Scholar
  24. 24.
    Weisstein, E.W.: Cograph. MathWorld, http://mathworld.wolfram.com/Cograph.html

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jean-François Couturier
    • 1
  • Pinar Heggernes
    • 2
  • Pim van ’t Hof
    • 2
  • Yngve Villanger
    • 2
  1. 1.LITAUniversité Paul Verlaine - MetzMetz Cedex 01France
  2. 2.Department of InformaticsUniversity of BergenBergenNorway

Personalised recommendations