ISCO 2018: Combinatorial Optimization pp 290-302

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

• Nina Chiarelli
• Tatiana Romina Hartinger
• Valeria Alejandra Leoni
• Maria Inés Lopez Pujato
• Martin Milanič
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).
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)
4. 4.
Bertossi, A.A.: Total domination in interval graphs. Inform. Process. Lett. 23(3), 131–134 (1986)
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)
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)
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)
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)
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).
10. 10.
Chang, G.J.: Labeling algorithms for domination problems in sun-free chordal graphs. Discrete Appl. Math. 22(1), 21–34 (1989)
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)
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)
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)
14. 14.
Chellali, M., Meddah, N.: Trees with equal 2-domination and 2-independence numbers. Discuss. Math. Graph Theory 32(2), 263–270 (2012)
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)
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)
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)
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)
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)
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)
21. 21.
Farber, M.: Independent domination in chordal graphs. Oper. Res. Lett. 1(4), 134–138 (1982)
22. 22.
Favaron, O., Hansberg, A., Volkmann, L.: On $$k$$-domination and minimum degree in graphs. J. Graph Theory 57(1), 33–40 (2008)
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)
25. 25.
Gerlach, T., Harant, J.: A note on domination in bipartite graphs. Discuss. Math. Graph Theory 22(2), 229–231 (2002)
26. 26.
Gutierrez, M., Oubiña, L.: Metric characterizations of proper interval graphs and tree-clique graphs. J. Graph Theory 21(2), 199–205 (1996)
27. 27.
Hansberg, A., Pepper, R.: On $$k$$-domination and $$j$$-independence in graphs. Discrete Appl. Math. 161(10–11), 1472–1480 (2013)
28. 28.
Harant, J., Pruchnewski, A., Voigt, M.: On dominating sets and independent sets of graphs. Combin. Probab. Comput. 8(6), 547–553 (1999)
29. 29.
Harary, F., Haynes, T.W.: Double domination in graphs. Ars Combin. 55, 201–213 (2000)
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)
33. 33.
Hsu, W.L., Tsai, K.-H.: Linear time algorithms on circular-arc graphs. Inform. Process. Lett. 40(3), 123–129 (1991)
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)
36. 36.
Jackowski, Z.: A new characterization of proper interval graphs. Discrete Math. 105(1–3), 103–109 (1992)
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)
39. 39.
Kazemi, A.P.: On the total $$k$$-domination number of graphs. Discuss. Math. Graph Theory 32(3), 419–426 (2012)
40. 40.
Keil, J.M.: Total domination in interval graphs. Inform. Process. Lett. 22(4), 171–174 (1986)
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)
43. 43.
Lan, J.K., Chang, G.J.: On the algorithmic complexity of $$k$$-tuple total domination. Discrete Appl. Math. 174, 81–91 (2014)
44. 44.
Lee, C.-M., Chang, M.-S.: Variations of $$Y$$-dominating functions on graphs. Discrete Math. 308(18), 4185–4204 (2008)
45. 45.
Liao, C.-S., Chang, G.J.: $$k$$-tuple domination in graphs. Inform. Process. Lett. 87(1), 45–50 (2003)
46. 46.
Liao, C.-S., Lee, D.T.: Power domination in circular-arc graphs. Algorithmica 65(2), 443–466 (2013)
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)
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)
49. 49.
Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Comput. Math. Appl. 25(7), 15–25 (1993)
50. 50.
Oum, S.-I., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theory Ser. B 96(4), 514–528 (2006)
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)
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)
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)
56. 56.
Ramalingam, G., Rangan, C.P.: A unified approach to domination problems on interval graphs. Inform. Process. Lett. 27(5), 271–274 (1988)
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)

© 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