A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs

  • Mamadou Moustapha KantéEmail author
  • Vincent Limouzy
  • Arnaud Mary
  • Lhouari Nourine
  • Takeaki Uno
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)


An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing the set of minimal transversals in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominating sets in chordal graphs, an important and well-studied graph class where such an algorithm was not known. The algorithm uses a new decomposition method of chordal graphs based on clique trees.


Maximal Clique Chordal Graph Extension Problem Graph Class Polynomial Space 
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.


  1. 1.
    Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., Verkamo, A.I.: Fast discovery of association rules. In: Advances in Knowledge Discovery and Data Mining, pp. 307–328. AAAI/MIT Press (1996)Google Scholar
  2. 2.
    Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: Dual-bounded generating problems: partial and multiple transversals of a hypergraph. SIAM J. Comput. 30(6), 2036–2050 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: Generating weighted transversals of a hypergraph. In: Rutgers University, pp. 13–22 (2000)Google Scholar
  4. 4.
    Courcelle, B.: Linear delay enumeration and monadic second-order logic. Discrete Appl. Math. 157(12), 2675–2700 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory (Graduate Texts in Mathematics). Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Dirac, G.A.: On rigid circuit graphs. Abhandlungen Aus Dem Mathematischen Seminare der Universität Hamburg 25(1–2), 71–76 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32(2), 514–537 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Galinier, P., Habib, M., Paul, C.: Chordal graphs and their clique graphs. In: Nagl, M. (ed.) WG 1995. LNCS, vol. 1017, pp. 358–371. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  11. 11.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theor. Ser. B 16(1), 47–56 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Golovach, P.A., Heggernes, P., Kante, M.M., Kratsch, D., Villanger, Y.: Enumerating minimal dominating sets in chordal bipartite graphs. Discrete Appl. Math. 199, 30–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gurvich, V.A.: On theory of multistep games. USSR Comput. Math. Math. Phys. 13(6), 143–161 (1973)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: On the neighbourhood helly of some graph classes and applications to the enumeration of minimal dominating sets. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 289–298. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: On the enumeration of minimal dominating sets and related notions. SIAM J. Discrete Math. 28(4), 1916–1929 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: On the enumeration and counting of minimal dominating sets in interval and permutation graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 339–349. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: Polynomial delay algorithm for listing minimal edge dominating sets in graphs. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 446–457. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  18. 18.
    Mary, A.: Énumeration des Dominants Minimaux d’un graphe. Ph.D. thesis, Université Blaise Pascal (2013)Google Scholar
  19. 19.
    Ramamurthy, K.G.: Coherent Structures and Simple Games. Theory and Decision Library. Game Theory, Mathematical Programming and Operations Research: Series C, vol. 6. Springer, Heidelberg (1990)CrossRefzbMATHGoogle Scholar
  20. 20.
    Strozecki, Y.: Enumeration Complexity and Matroid Decomposition. Ph.D. thesis, Université Paris Diderot - Paris 7 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Mamadou Moustapha Kanté
    • 1
    Email author
  • Vincent Limouzy
    • 1
  • Arnaud Mary
    • 2
  • Lhouari Nourine
    • 1
  • Takeaki Uno
    • 3
  1. 1.Clermont-Université, Université Blaise Pascal, LIMOS, CNRSClermont-FerrandFrance
  2. 2.Université Claude Bernard Lyon 1, LBBE, CNRSVilleurbanneFrance
  3. 3.National Institute of InformaticsTokyoJapan

Personalised recommendations