Efficient Domination for Some Subclasses of \(P_6\)-free Graphs in Polynomial Time

  • Andreas Brandstädt
  • Elaine M. Eschen
  • Erik FrieseEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)


Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be \(\mathbb {NP}\)-complete even for very restricted graph classes such as \(P_7\)-free chordal graphs. The ED problem on a graph G can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of G. The complexity of the ED problem is an open question for \(P_6\)-free graphs and was open even for the subclass of \(P_6\)-free chordal graphs. In this paper, we show that squares of \(P_6\)-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (\(P_6\), house, hole, domino)-free graphs. This implies that ED/WeightedED is solvable in polynomial time for (\(P_6\), house, hole, domino)-free graphs; in particular, for \(P_6\)-free chordal graphs. Moreover, based on our result that squares of \(P_6\)-free graphs that have an e.d. are hole-free and some properties concerning odd antiholes, we show that squares of (\(P_6\), house)-free graphs ((\(P_6\), bull)-free graphs, respectively) that have an e.d. are perfect. This implies that ED/WeightedED is solvable in polynomial time for (\(P_6\), house)-free graphs and for (\(P_6\), bull)-free graphs (the time bound for (\(P_6\), house, hole, domino)-free graphs is better than that for (\(P_6\), house)-free graphs). The complexity of the ED problem for \(P_6\)-free graphs remains an open question.


Efficient domination Chordal graphs Hole-free graphs (house, hole, domino)-free graphs \(P_6\)-free graphs Polynomial-time algorithm 



The first and second authors gratefully acknowledge support from the West Virginia University NSF ADVANCE Sponsorship Program, and the first author thanks Van Bang Le for discussions about the Efficient Domination problem.


  1. 1.
    Brandstädt, A., Eschen, E.M., Friese, E.: Efficient domination for some subclasses of \(P_6\)-free graphs in polynomial time, arXiv:1503.00091v1 (2015)
  2. 2.
    Brandstädt, A., Fičur, P., Leitert, A., Milanič, M.: Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs. Inf. Process. Lett. 115, 256–262 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Giakoumakis, V.: Weighted efficient domination for \((P_5+kP_2)\)-free graphs in polynomial time, arXiv:1407.4593v1 (2014)
  4. 4.
    Brandstädt, A., Karthick, T., Weighted efficient domination in classes of \(P_6\)-free graphs, arXiv:1503.06025v1 (2015)
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. In: SIAM Monographs on Discrete Mathematics Application, vol. 3. SIAM, Philadelphia (1999)Google Scholar
  6. 6.
    Brandstädt, A., Leitert, A., Rautenbach, D.: Efficient dominating and edge dominating sets for graphs and hypergraphs. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 267–277. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Brandstädt, A., Milanič, M., Nevries, R.: New polynomial cases of the weighted efficient domination problem. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 195–206. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frank, A.: Some polynomial algorithms for certain graphs and hypergraphs. In: Proceedings of 5th British Combinatorial Conference 1976, Aberdeen, Congressus Numerantium No. XV, pp. 211–226 (1975)Google Scholar
  10. 10.
    Friese, E.: Das efficient-domination-problem auf \(P_6\)-freien graphen, Master Thesis. University of Rostock, Germany (2013) (in German)Google Scholar
  11. 11.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karthick, T.: Weighted Efficient Domination for Certain Classes of \(P_6\)-free Graphs, Manuscript (2015)Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability - A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  14. 14.
    Leitert, A.: Das dominating induced matching problem für azyklische hypergraphen, Diploma Thesis. University of Rostock, Germany (2012) (in German)Google Scholar
  15. 15.
    Lokshtanov, D., Pilipczuk, M., van Leeuwen, E.J.: Independence and efficient domination on P6-free graphs, arXiv:1507.02163v1 (2015)
  16. 16.
    Milanič, M.: Hereditary efficiently dominatable graphs. J. Graph Theor. 73, 400–424 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Andreas Brandstädt
    • 1
  • Elaine M. Eschen
    • 2
  • Erik Friese
    • 3
    Email author
  1. 1.Institut für InformatikUniversität RostockRostockGermany
  2. 2.West Virginia UniversityMorgantownUSA
  3. 3.Institut für MathematikUniversität RostockRostockGermany

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