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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.

Cite AsGet BibTex

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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