, Volume 68, Issue 4, pp 998–1018 | Cite as

Dominating Induced Matchings for P 7-Free Graphs in Linear Time

  • Andreas BrandstädtEmail author
  • Raffaele Mosca


Let G be a finite undirected graph with edge set E. An edge set E′⊆E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge eEE′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem.

The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is \({\mathbb{NP}}\)-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for P k -free graphs for any k≥5; P k denotes a chordless path with k vertices and k−1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P 7-free graphs in a robust way.


Dominating induced matching Efficient edge domination P7-free graphs Linear time algorithm Robust algorithm 



The first author gratefully acknowledges a research stay at the LIMOS institute, University of Clermont-Ferrand, and the inspiring discussions with Anne Berry on dominating induced matchings.


  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974) zbMATHGoogle Scholar
  2. 2.
    Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory 41, 182–208 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L., Brandstädt, A., Kratsch, D., Rao, M., Spinrad, J.: On algorithms for (P 5, gem)-free graphs. Theor. Comput. Sci. 349, 2–21 (2005) CrossRefzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Hundt, C., Nevries, R.: Efficient Edge Domination on Hole-Free graphs. In: Polynomial Time, Conference Proceedings LATIN 2010. Lecture Notes in Computer Science, vol. 6034, pp. 650–661 (2010) Google Scholar
  5. 5.
    Brandstädt, A., Kratsch, D.: On the structure of (P 5, gem)-free graphs. Discrete Appl. Math. 145, 155–166 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brandstädt, A., Le, H.-O., Mosca, R.: Chordal co-gem-free and (P 5, gem)-free graphs have bounded clique-width. Discrete Appl. Math. 145, 232–241 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Brandstädt, A., Leitert, A., Rautenbach, D.: Efficient dominating and edge dominating sets for graphs and hypergraphs (2012). Extended abstract accepted for ISAAC 2012, Taiwan. arXiv:1207.0953v2 [cs.DM]
  8. 8.
    Bretscher, A., Corneil, D.G., Habib, M., Paul, Ch.: A simple linear time LexBFS cograph recognition algorithm. SIAM J. Discrete Math. 22(4), 1277–1296 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cardoso, D.M., Korpelainen, N., Lozin, V.V.: On the complexity of the dominating induced matching problem in hereditary classes of graphs. Discrete Appl. Math. 159, 521–531 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cardoso, D.M., Lozin, V.V.: Dominating induced matchings. In: “Graph Theory, Computational Intelligence and Thought”, A Conference Celebrating Marty Golumbic’s 60th Birthday, Jerusalem, Tiberias, Haifa, 2008. Lecture Notes in Computer Science, vol. 5420, pp. 77–86 (2009) CrossRefGoogle Scholar
  11. 11.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14, 926–934 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique width. Theory Comput. Syst. 33, 125–150 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci. 11, 423–443 (2000) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Grinstead, D.L., Slater, P.L., Sherwani, N.A., Holmes, N.D.: Efficient edge domination problems in graphs. Inf. Process. Lett. 48, 221–228 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hopcroft, J.E., Tarjan, R.E.: Efficient algorithms for graph manipulation [H]. Commun. ACM 16(6), 372–378 (1973) CrossRefGoogle Scholar
  16. 16.
    Livingston, M., Stout, Q.: Distributing resources in hypercube computers. In: Proceedings 3rd Conf. on Hypercube Concurrent Computers and Applications, pp. 222–231 (1988) Google Scholar
  17. 17.
    Lu, C.L., Ko, M.-T., Tang, C.Y.: Perfect edge domination and efficient edge domination in graphs. Discrete Appl. Math. 119(3), 227–250 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lu, C.L., Tang, C.Y.: Efficient domination in bipartite graphs. Manuscript (1997) Google Scholar
  19. 19.
    McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Math. 201, 189–241 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Spinrad, J.P.: Efficient Graph Representations. Fields Institute Monographs. Am. Math. Soc., Providence (2003) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Fachbereich InformatikUniversität RostockRostockGermany
  2. 2.Dipartimento di ScienzeUniversitá degli Studi “G. D’Annunzio”PescaraItaly

Personalised recommendations