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Algorithmic Aspects of Disjunctive Total Domination in Graphs

  • Chin-Fu Lin
  • Sheng-Lung Peng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)

Abstract

For a graph \(G=(V,E)\), \(D\subseteq V\) is a dominating set if every vertex in \(V\!\setminus \!D\) has a neighbor in D. If every vertex in V has to be adjacent to a vertex of D, then D is called a total dominating set of G. The (total) domination problem on G is to find a (total) dominating set D of the minimum cardinality. The (total) domination problem is well-studied. Recently, the following variant is proposed. Vertex subset D is a disjunctive total dominating set if every vertex of V is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The disjunctive total domination problem on G is to find a disjunctive total dominating set D of the minimum cardinality. For the complexity issue, the only known result is that the disjunctive total domination problem is NP-hard on general graphs. In this paper, by using a minimum-cost flow algorithm as a subroutine, we show that the disjunctive total domination problem on trees can be solved in polynomial time. This is the first polynomial-time algorithm for the problem on a special class of graphs. Besides, we show that the problem remains NP-hard on bipartite graphs and planar graphs.

Keywords

Trees Total domination Disjunctive total domination Minimum-cost flow algorithm 

Notes

Acknowledgement

This work was partially supported by the Ministry of Science and Technology of Taiwan, under Contract No. MOST 105-2221-E-259-018.

References

  1. 1.
    Booth, K.S., Johnson, J.H.: Dominating sets in chordal graphs. SIAM J. Comput. 11, 191–199 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chang, G.J.: Labeling algorithms for domination problems in sun-free chordal graphs. Discrete Appl. Math. 22, 21–34 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  4. 4.
    Goddard, W., Henning, M.A., McPillan, C.A.: The disjunctive domination number of a graph. Quaestiones Math. 37, 547–561 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Henning, M.A.: Graphs with large total domination number. J. Graph Theor. 35, 21–45 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Henning, M.A.: A survey of selected recent results on total domination in graphs. Discrete Math. 309, 32–63 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Henning, M.A., Marcon, S.A.: Domination versus disjunctive domination in trees. Discrete Appl. Math. 184, 171–177 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Henning, M.A., Marcon, S.A.: Domination versus disjunctive domination in graphs. Quaestiones Math. 39, 261–273 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Henning, M.A., Naicker, V.: Graphs with large disjunctive total domination number. Discrete Math. Theor. Comput. Sci. 17, 255–282 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Henning, M.A., Naicker, V.: Bounds on the disjunctive total domination number of a tree. Discussiones Math. Graph Theor. 36, 153–171 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Henning, M.A., Naicker, V.: Disjunctive total domination in graphs. J. Comb. Optim. 31, 1090–1110 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Henning, M.A., Yeo, A.: Total Domination in Graphs. Springer, New York (2013). ISBN:978-1-4614-6525-6CrossRefzbMATHGoogle Scholar
  13. 13.
    Keil, J.M.: The complexity of domination problems in circle graphs. Discrete Appl. Math. 42, 51–63 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kovács, P.: Minimum-cost flow algorithms: an experimental evaluation. Optim. Methods Softw. 30, 94–127 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kratsch, D., Stewart, L.: Total domination and transformation. Inf. Process. Lett. 63, 167–170 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laskar, R., Pfaff, J., Hedetniemi, S.M., Hedetneimi, S.T.: On the algorithmic complexity of total domination. SIAM. J. Algebraic Discrete Methods 5, 420–425 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Müller, H., Brandstädt, A.: The NP-completeness of steiner tree and dominating set for chordal bipartite graphs. Theor. Comput. Sci. 53, 257–265 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Oper. Res. 41, 338–350 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Panda, B.S., Pandey, A., Paul, S.: Algorithmic aspects of disjunctive domination in graphs. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 325–336. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21398-9_26 CrossRefGoogle Scholar
  20. 20.
    Pfaff, J., Laskar, R.C., Hedetniemi, S.T.: NP-completeness of total and connected domination and irredundance for bipartite graphs. Technical report 428, Clemson University. Dept. Math. Sciences (1983)Google Scholar
  21. 21.
    Ramalingam, G., Pandu Rangan, C.: Total domination in interval graphs revisited. Inf. Process. Lett. 27, 17–21 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Telle, J.A.: Complexity of domination-type problems in graphs. Nord. J. Comput. 1, 157–171 (1994)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Computer Science and Information EngineeringNational Dong Hwa UniversityHualienTaiwan

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