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Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P_7 - Free and P_8 - Free Chordal Graphs

Authors Oscar Defrain, Lhouari Nourine

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Oscar Defrain
  • Université Clermont Auvergne, France
  • oscar.defrain@uca.fr
Lhouari Nourine
  • Université Clermont Auvergne, France
  • lhouari.nourine@uca.fr

Acknowledgements

The authors are supported by the ANR project GraphEn ANR-15-CE40-0009.

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Oscar Defrain and Lhouari Nourine. Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P_7 - Free and P_8 - Free Chordal Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ISAAC.2019.63

Abstract

In [M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal on Discrete Mathematics, 28(4):1916–1929, 2014.] the authors give an O(n+m) delay algorithm based on neighborhood inclusions for the enumeration of minimal dominating sets in split and P_6-free chordal graphs. In this paper, we investigate generalizations of this technique to P_k-free chordal graphs for larger integers k. In particular, we give O(n+m) and O(n^3 * m) delays algorithms in the classes of P_7-free and P_8-free chordal graphs. As for P_k-free chordal graphs for k >= 9, we give evidence that such a technique is inefficient as a key step of the algorithm, namely the irredundant extension problem, becomes NP-complete.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph enumeration
Keywords
  • Minimal dominating sets
  • enumeration algorithms
  • linear delay enumeration
  • chordal graphs
  • forbidden induced paths

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References

  1. Hiroki Arimura and Takeaki Uno. Polynomial-delay and polynomial-space algorithms for mining closed sequences, graphs, and pictures in accessible set systems. In Proceedings of the 2009 SIAM International Conference on Data Mining, pages 1088-1099. SIAM, 2009. Google Scholar
  2. Marthe Bonamy, Oscar Defrain, Marc Heinrich, Michał Pilipczuk, and Jean-Florent Raymond. Enumerating minimal dominating sets in K_t-free graphs and variants. arXiv preprint, 2019. URL: http://arxiv.org/abs/1810.00789.
  3. Marthe Bonamy, Oscar Defrain, Marc Heinrich, and Jean-Florent Raymond. Enumerating Minimal Dominating Sets in Triangle-Free Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science, pages 16:1-16:12. Springer, 2019. Google Scholar
  4. Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Leonid Khachiyan. Generating maximal independent sets for hypergraphs with bounded edge-intersections. In Latin American Symposium on Theoretical Informatics, pages 488-498. Springer, 2004. Google Scholar
  5. Bruno Courcelle. Linear delay enumeration and monadic second-order logic. Discrete Applied Mathematics, 157(12):2675-2700, 2009. Google Scholar
  6. Nadia Creignou, Markus Kröll, Reinhard Pichler, Sebastian Skritek, and Heribert Vollmer. A complexity theory for hard enumeration problems. Discrete Applied Mathematics, 2019. Google Scholar
  7. Thomas Eiter, Georg Gottlob, and Kazuhisa Makino. New results on monotone dualization and generating hypergraph transversals. SIAM Journal on Computing, 32(2):514-537, 2003. Google Scholar
  8. Michael L. Fredman and Leonid Khachiyan. On the Complexity of Dualization of Monotone Disjunctive Normal Forms. Journal of Algorithms, 21(3):618-628, 1996. Google Scholar
  9. Petr A. Golovach, Pinar Heggernes, Mamadou M. Kanté, Dieter Kratsch, and Yngve Villanger. Enumerating minimal dominating sets in chordal bipartite graphs. Discrete Applied Mathematics, 199:30-36, 2016. Google Scholar
  10. David S. Johnson, Mihalis Yannakakis, and Christos H. Papadimitriou. On generating all maximal independent sets. Information Processing Letters, 27(3):119-123, 1988. Google Scholar
  11. Mamadou M. Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. On the neighbourhood helly of some graph classes and applications to the enumeration of minimal dominating sets. In International Symposium on Algorithms and Computation, pages 289-298. Springer, 2012. Google Scholar
  12. Mamadou M. Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. On the Enumeration of Minimal Dominating Sets and Related Notions. SIAM Journal on Discrete Mathematics, 28(4):1916-1929, 2014. Google Scholar
  13. Mamadou M. Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. A polynomial delay algorithm for enumerating minimal dominating sets in chordal graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 138-153. Springer, 2015. Google Scholar
  14. Mamadou M. Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. Polynomial delay algorithm for listing minimal edge dominating sets in graphs. In Workshop on Algorithms and Data Structures, pages 446-457. Springer, 2015. Google Scholar
  15. Ronald C. Read and Robert E. Tarjan. Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks, 5(3):237-252, 1975. Google Scholar
  16. Yann Strozecki and Arnaud Mary. Efficient enumeration of solutions produced by closure operations. Discrete Mathematics & Theoretical Computer Science, 21, 2019. Google Scholar
  17. Elliot S. Wolk. The comparability graph of a tree. Proceedings of the American Mathematical Society, 13(5):789-795, 1962. Google Scholar
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