, Volume 80, Issue 2, pp 714–741 | Cite as

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

  • Petr A. Golovach
  • Pinar Heggernes
  • Mamadou Moustapha Kanté
  • Dieter Kratsch
  • Sigve H. Sæther
  • Yngve Villanger


The linear induced matching width (LMIM-width) of a graph is a width parameter defined by using the notion of branch-decompositions of a set function on ternary trees. In this paper we study output-polynomial enumeration algorithms on graphs of bounded LMIM-width and graphs of bounded local LMIM-width. In particular, we show that all 1-minimal and all 1-maximal \((\sigma ,\rho )\)-dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear) delay using polynomial space. Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.


Domination problem Local linear MIM-width Output-polynomial enumeration Linear delay 


  1. 1.
    Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1–3), 21–46 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidimensional parameters and local treewidth. SIAM J. Discrete Math. 18(3), 501–511 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Demaine, E.D., Hajiaghayi, M., Thilikos, D.M.: The bidimensional theory of bounded-genus graphs. SIAM J. Discrete Math. 20(2), 357–371 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Belmonte, R., Vatshelle, M.: Graph classes with structured neighborhoods and algorithmic applications. Theor. Comput. Sci. 511, 54–65 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. In: SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)Google Scholar
  6. 6.
    Breu, H.: Algorithmic Aspects of Constrained Unit Disk Graphs. Ph.D. thesis, The University of British Columbia (1996)Google Scholar
  7. 7.
    Bui-Xuan, B.M., Telle, J.A., Vatshelle, M.: Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theor. Comput. Sci. 511, 66–76 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chandran, L.S., Francis, M.C., Sivadasan, N.: On the cubicity of interval graphs. Graphs Comb. 25(2), 169–179 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Courcelle, B.: Linear delay enumeration and monadic second-order logic. Discrete Appl. Math. 157, 2675–2700 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24, 1278–1304 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eiter, T., Gottlob, G.: Hypergraph transversal computation and related problems in logic and AI. In: Proceedings of the JELIA 2002. LNCS, vol. 2424, pp. 549–564 (2002)Google Scholar
  12. 12.
    Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32, 514–537 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Golovach, P.A., Heggernes, P., Kratsch, D., Villanger, Y.: An incremental polynomial time algorithm to enumerate all minimal edge dominating sets. Algorithmica 72, 836–859 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Golovach, P.A., Heggernes, P., Kanté, M.M., Kratsch, D., Villanger, Y.: Enumerating minimal dominating sets in chordal bipartite graphs. Discrete Appl. Math. 166, 30–35 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: On the enumeration of minimal dominating sets and related notions. SIAM J. Discrete Math. 28, 1916–1929 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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: Proceedings of the ISAAC 2012. LNCS, vol. 7676, pp. 289–298 (2012)Google Scholar
  18. 18.
    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: Proceedings of the ISAAC 2013. LNCS, vol. 8283, pp. 339–349 (2013)Google Scholar
  19. 19.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: A polynomial delay algorithm for enumerating minimal dominating sets in chordal graphs. In: Proceedings of the WG 2015. LNCS, vol. 9224, pp. 138–153 (2015)Google Scholar
  20. 20.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: Polynomial delay algorithm for listing minimal edge dominating sets in graphs. In: Proceedings of the WADS 2015, LNCS, vol. 9214, pp. 446–357 (2015)Google Scholar
  21. 21.
    Khachiyan, L., Boros, E., Borys, K., Elbassioni, K.M., Gurvich, V.: Generating all vertices of a polyhedron is hard. Discrete Comput. Geom. 39, 174–190 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Khachiyan, L., Boros, E., Elbassioni, K.M., Gurvich, V.: On enumerating minimal dicuts and strongly connected subgraphs. Algorithmica 50, 159–172 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets. SIAM J. Comput. 9, 558–565 (1980)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rauf, I.: Polynomially Solvable Cases of Hypergraph Transversal and Related Problems. Ph.D. thesis, Saarland University (2011)Google Scholar
  25. 25.
    Tarjan, R.E.: Enumeration of the elementary circuits of a directed graph. SIAM J. Comput. 2, 211–216 (1973)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial \(k\)-trees. SIAM J. Discrete Math. 10(4), 529–550 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Vatshelle, M.: New Width Parameters of Graphs. Ph.D. thesis, University of Bergen (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Pinar Heggernes
    • 1
  • Mamadou Moustapha Kanté
    • 2
  • Dieter Kratsch
    • 3
  • Sigve H. Sæther
    • 1
  • Yngve Villanger
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.LIMOS, CNRSUniversité Clermont AuvergneAubiéreFrance
  3. 3.LITAUniversité de LorraineMetzFrance

Personalised recommendations