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Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

  • Nina Chiarelli
  • Tatiana Romina Hartinger
  • Valeria Alejandra LeoniEmail author
  • Maria Inés Lopez Pujato
  • Martin MilaničEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10856)

Abstract

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-dominating set, in a given graph, are referred to as k-domination, resp. total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work by Kang et al. (2017) that these two families of problems are solvable in time \(\mathcal {O}(|V(G)|^{6k+4})\) in the class of interval graphs. In this work, we develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs. The algorithms run in time \(\mathcal {O}(|V(G)|^{3k})\) for each fixed \(k\ge 1\) and are also applicable to the weighted case.

Keywords

k-domination Total k-domination Proper interval graph Polynomial-time algorithm 

References

  1. 1.
    Argiroffo, G., Leoni, V., Torres, P.: Complexity of \(k\)-tuple total and total \(\{k\}\)-dominations for some subclasses of bipartite graphs Inform. Process. Lett. (2017).  https://doi.org/10.1016/j.ipl.2018.06.007
  2. 2.
    Bakhshesh, D., Farshi, M., Hasheminezhad, M.: Complexity results for \(k\)-domination and \(\alpha \)-domination problems and their variants (2017). arXiv:1702.00533 [cs.CC]
  3. 3.
    Belmonte, R., Vatshelle, M.: Graph classes with structured neighborhoods and algorithmic applications. Theoret. Comput. Sci. 511, 54–65 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertossi, A.A.: Total domination in interval graphs. Inform. Process. Lett. 23(3), 131–134 (1986)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Braga, A., de Souza, C.C., Lee, O.: The eternal dominating set problem for proper interval graphs. Inform. Process. Lett. 115(6–8), 582–587 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brandstädt, A., Chepoi, V.D., Dragan, F.F.: The algorithmic use of hypertree structure and maximum neighbourhood orderings. Discrete Appl. Math. 82(1–3), 43–77 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brešar, B., Gologranc, T., Kos, T.: Dominating sequences under atomic changes with applications in sierpiński and interval graphs. Appl. Anal. Discrete Math. 10(2), 518–531 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoret. Comput. Sci. 511, 66–76 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cattanéo, D., Perdrix, S.: The parameterized complexity of domination-type problems and application to linear codes. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds.) TAMC 2014. LNCS, vol. 8402, pp. 86–103. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-06089-7_7CrossRefzbMATHGoogle Scholar
  10. 10.
    Chang, G.J.: Labeling algorithms for domination problems in sun-free chordal graphs. Discrete Appl. Math. 22(1), 21–34 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chang, G.J., Pandu, C.P., Coorg, S.R.: Weighted independent perfect domination on cocomparability graphs. Discrete Appl. Math. 63(3), 215–222 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chang, M.-S.: Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput. 27(6), 1671–1694 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chellali, M., Favaron, O., Hansberg, A., Volkmann, L.: \(k\)-domination and \(k\)-independence in graphs: a survey. Graphs Combin. 28(1), 1–55 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chellali, M., Meddah, N.: Trees with equal 2-domination and 2-independence numbers. Discuss. Math. Graph Theory 32(2), 263–270 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cheng, T.C.E., Kang, L.Y., Ng, C.T.: Paired domination on interval and circular-arc graphs. Discrete Appl. Math. 155(16), 2077–2086 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cicalese, F., Cordasco, G., Gargano, L., Milanič, M., Vaccaro, U.: Latency-bounded target set selection in social networks. Theoret. Comput. Sci. 535, 1–15 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cicalese, F., Milanič, M., Vaccaro, U.: On the approximability and exact algorithms for vector domination and related problems in graphs. Discrete Appl. Math. 161(6), 750–767 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Corneil, D.G.: A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs. Discrete Appl. Math. 138(3), 371–379 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    DeLaViña, E., Goddard, W., Henning, M.A., Pepper, R., Vaughan, E.R.: Bounds on the \(k\)-domination number of a graph. Appl. Math. Lett. 24(6), 996–998 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Farber, M.: Independent domination in chordal graphs. Oper. Res. Lett. 1(4), 134–138 (1982)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Favaron, O., Hansberg, A., Volkmann, L.: On \(k\)-domination and minimum degree in graphs. J. Graph Theory 57(1), 33–40 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fink, J.F., Jacobson, M.S.: \(n\)-domination in graphs. In: Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984), pp. 283–300. Wiley, New York (1985)Google Scholar
  24. 24.
    Gardi, F.: The roberts characterization of proper and unit interval graphs. Discrete Math. 307(22), 2906–2908 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gerlach, T., Harant, J.: A note on domination in bipartite graphs. Discuss. Math. Graph Theory 22(2), 229–231 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Gutierrez, M., Oubiña, L.: Metric characterizations of proper interval graphs and tree-clique graphs. J. Graph Theory 21(2), 199–205 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hansberg, A., Pepper, R.: On \(k\)-domination and \(j\)-independence in graphs. Discrete Appl. Math. 161(10–11), 1472–1480 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Harant, J., Pruchnewski, A., Voigt, M.: On dominating sets and independent sets of graphs. Combin. Probab. Comput. 8(6), 547–553 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Harary, F., Haynes, T.W.: Double domination in graphs. Ars Combin. 55, 201–213 (2000)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Domination in Graphs. Advanced Topics, volume 209 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (1998)Google Scholar
  31. 31.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of domination in graphs. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 208. Marcel Dekker Inc., New York (1998)Google Scholar
  32. 32.
    Henning, M.A., Kazemi, A.P.: \(k\)-tuple total domination in graphs. Discrete Appl. Math. 158(9), 1006–1011 (2010)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Hsu, W.L., Tsai, K.-H.: Linear time algorithms on circular-arc graphs. Inform. Process. Lett. 40(3), 123–129 (1991)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ishii, T., Ono, H., Uno, Y.: Subexponential fixed-parameter algorithms for partial vector domination. Discrete Optim. 22(part A), 111–121 (2016)Google Scholar
  35. 35.
    Ishii, T., Ono, H., Uno, Y.: (Total) vector domination for graphs with bounded branchwidth. Discrete Appl. Math. 207, 80–89 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Jackowski, Z.: A new characterization of proper interval graphs. Discrete Math. 105(1–3), 103–109 (1992)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Jacobson, M.S., Peters, K.: Complexity questions for \(n\)-domination and related parameters. Congr. Numer. 68, 7–22 (1989). Eighteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1988)Google Scholar
  38. 38.
    Kang, D.Y., Kwon, O.-J., Strømme, T.J.F., Telle, J.A.: A width parameter useful for chordal and co-comparability graphs. Theoret. Comput. Sci. 704, 1–17 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kazemi, A.P.: On the total \(k\)-domination number of graphs. Discuss. Math. Graph Theory 32(3), 419–426 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Keil, J.M.: Total domination in interval graphs. Inform. Process. Lett. 22(4), 171–174 (1986)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Kulli, V.R.: On \(n\)-total domination number in graphs. Graph Theory. Combinatorics, Algorithms, and Applications (San Francisco, CA, 1989), pp. 319–324. SIAM, Philadelphia (1991)Google Scholar
  42. 42.
    Lan, J.K., Chang, G.J.: Algorithmic aspects of the \(k\)-domination problem in graphs. Discrete Appl. Math. 161(10–11), 1513–1520 (2013)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Lan, J.K., Chang, G.J.: On the algorithmic complexity of \(k\)-tuple total domination. Discrete Appl. Math. 174, 81–91 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Lee, C.-M., Chang, M.-S.: Variations of \(Y\)-dominating functions on graphs. Discrete Math. 308(18), 4185–4204 (2008)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Liao, C.-S., Chang, G.J.: \(k\)-tuple domination in graphs. Inform. Process. Lett. 87(1), 45–50 (2003)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Liao, C.-S., Lee, D.T.: Power domination in circular-arc graphs. Algorithmica 65(2), 443–466 (2013)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Liedloff, M., Kloks, T., Liu, J., Peng, S.-L.: Efficient algorithms for Roman domination on some classes of graphs. Discrete Appl. Math. 156(18), 3400–3415 (2008)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Lin, C.-J., Liu, J.-J., Wang, Y.-L.: Finding outer-connected dominating sets in interval graphs. Inform. Process. Lett. 115(12), 917–922 (2015)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Comput. Math. Appl. 25(7), 15–25 (1993)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Oum, S.-I., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theory Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Panda, B.S., Paul, S.: A linear time algorithm for liar’s domination problem in proper interval graphs. Inform. Process. Lett. 113(19–21), 815–822 (2013)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Panda, B.S., Pradhan, D.: A linear time algorithm to compute a minimum restrained dominating set in proper interval graphs. Discrete Math. Algorithms Appl. 7(2), 1550020 (2015). 21Google Scholar
  53. 53.
    Pradhan, D.: Algorithmic aspects of \(k\)-tuple total domination in graphs. Inform. Process. Lett. 112(21), 816–822 (2012)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Pramanik, T., Mondal, S., Pal, M.: Minimum \(2\)-tuple dominating set of an interval graph. Int. J. Comb. 14 (2011). Article ID 389369Google Scholar
  55. 55.
    Ramalingam, G., Pandu, C.: Total domination in interval graphs revisited. Inform. Process. Lett. 27(1), 17–21 (1988)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Ramalingam, G., Rangan, C.P.: A unified approach to domination problems on interval graphs. Inform. Process. Lett. 27(5), 271–274 (1988)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Roberts, F.S.: Indifference graphs. In: Proof Techniques in Graph Theory (Proceedings of Second Ann Arbor Graph Theory Conference, Ann Arbor, Michigan, 1968), pp. 139–146. Academic Press, New York (1969)Google Scholar
  58. 58.
    Chiarelli, N., Hartinger, T.R., Leoni, V.A., Lopez Pujato, M.I., Milanič, M.: New algorithms for weighted \(k\)-domination and total \(k\)-domination problems in proper interval graphs. arXiv:1803.04327 [cs.DS] (2018)

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Nina Chiarelli
    • 1
    • 2
  • Tatiana Romina Hartinger
    • 1
    • 2
  • Valeria Alejandra Leoni
    • 3
    • 4
    Email author
  • Maria Inés Lopez Pujato
    • 3
    • 5
  • Martin Milanič
    • 1
    • 2
    Email author
  1. 1.FAMNITUniversity of PrimorskaKoperSlovenia
  2. 2.IAMUniversity of PrimorskaKoperSlovenia
  3. 3.FCEIAUniversidad Nacional de RosarioRosarioArgentina
  4. 4.CONICETBuenos AiresArgentina
  5. 5.ANPCyTBuenos AiresArgentina

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