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Minimum 2-tuple dominating set of permutation graphs

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Abstract

For a fixed positive integer k, a k-tuple dominating set of a graph G=(V,E) is a subset D⊆V such that every vertex in V is dominated by at least k vertex in D. The k-tuple domination number γ ×k (G) is the minimum size of a k-tuple dominating set of G. The special case when k=1 is the usual domination. The case when k=2 was called double domination or 2-tuple domination. A 2-tuple dominating set D 2 is said to be minimal if there does not exist any D′⊂D 2 such that D′ is a 2-tuple dominating set of G. A 2-tuple dominating set D 2, denoted by γ ×2(G), is said to be minimum, if it is minimal as well as it gives 2-tuple domination number. In this paper, we present an efficient algorithm to find a minimum 2-tuple dominating set on permutation graphs with n vertices which runs in O(n 2) time.

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References

  1. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  2. Awerbuch, B., Peleg, D.: Sparse partitions (extended abstract). In: 31st Annual Symposium on Foundations of Computer Science, pp. 503–513 (1990)

    Chapter  Google Scholar 

  3. Benson, D.A., Boguski, M.S., Lipman, D.J., Ostell, J., Ouellette, B.F., Rapp, B.A., Wheeler, D.L.: Nucleic acids research. GenBank 27(1), 12–17 (1999)

    Google Scholar 

  4. Berchtold, S., Keim, D., Kriegel, H.-P.: Transitive orientation of graphs and identification of permutation graphs. In: Proceedings of the 22nd Conference on Very Large Databases, pp. 28–39 (1996)

    Google Scholar 

  5. Berchtold, S., Bohm, C., Kriegel, H.P.: The pyramid-technique: towards breaking the curse of dimensionality. In: Proceedings of the SIGMOD Conference, pp. 142–153 (1998)

    Google Scholar 

  6. Berger, E.: Dynamic monopolies of constant size. Technical report, Los Alamos National Laboratories (1999)

  7. Chang, G.J.: Algorithmic aspect of domination in graphs. In: Du, D.-Z., Paradolas, P.M. (eds.) Handbook of Combinatorial Optimization, vol. 3, pp. 339–405 (1998)

    Google Scholar 

  8. Chao, H.S., Hsu, F.R., Lee, R.C.T.: An optimal algorithm for finding the minimum cardinality dominating set on permutation graphs. Discrete Appl. Math. 102, 159–173 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockayne, E., Dawes, R., Hedetniemi, S.: Total domination in graphs. Networks 10, 211–219 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. De Jaenisch, C.F.: Trait des Applications de l’Analyse Mathematique au Jeu des Echecs. Petrograd (1862)

  11. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  12. Harary, F., Haynes, T.W.: Double domination in graphs. Ars Comb. 55, 201–213 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Haynes, T.W., Slater, P.J.: Paired-domination in graphs. Networks 32, 199–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Dekker, New York (1998)

    MATH  Google Scholar 

  15. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Selected Topics. Dekker, New York (1998)

    Google Scholar 

  16. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: The Theory. Dekker, New York (1998)

    MATH  Google Scholar 

  17. Haynes, T.W., Mynhardt, C.M., van der Merwe, L.C.: Criticality index of total domination. Congr. Numer. 131, 67–73 (1998)

    MathSciNet  MATH  Google Scholar 

  18. Haynes, T.W., Mynhardt, C.M., van der Merwe, L.C.: Total domination edge critical graphs. Util. Math. 54, 229–240 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Dekker, New York (1998)

    MATH  Google Scholar 

  20. Haynes, T.W., Mynhardt, C.M., van der Merwe, L.C.: Total domination edge critical graphs with maximum diameter. Discuss. Math., Graph Theory 21, 187–205 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liao, C.S., Chang, G.J.: Algorithmic aspect of k-tuple domination in graphs. Taiwan. J. Math. 6, 415–420 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Liao, C.S., Chang, G.J.: k-Tuple domination in graphs. Inf. Process. Lett. 87, 45–50 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, C.L., Ko, M.-T., Tang, C.Y.: Perfect edge domination and efficient edge domination in graphs. Discrete Appl. Math. 119, 227–250 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Muller, J.H., Spinrad, J.: Incremental modular decomposition. J. ACM 36, 1–19 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pnueli, A., Lempel, A., Even, S.: Transitive orientation of graphs and identification of permutation graphs. Can. J. Math. 23, 160–175 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pramanik, T., Mondal, S., Pal, M.: Minimum 2-tuple dominating set of an interval graphs. Int. J. Comb. 2011, 1–14 (2011)

    MathSciNet  Google Scholar 

  27. Qiao, H., Kang, L., Gardei, M., Du, D.-Z.: Paired-domination of trees. J. Glob. Optim. 25, 43–54 (2003)

    Article  MATH  Google Scholar 

  28. Shan, E., Dong, Y.: The k-tuple twin domination in generalized de Bruijn and Kautz networks. Comput. Math. Appl. 63(1), 222–227 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sumner, D.P., Blitch, P.: Domination critical graphs. J. Comb. Theory, Ser. B 34, 65–76 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yannakakis, M., Gavril, F.: Edge domination sets in graphs. SIAM J. Appl. Math. B 38, 364–372 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721 102, India

    Sambhu Charan Barman & Madhumangal Pal

  2. Department of Mathematics, Raja N.L. Khan Women’s College, Gope Palace, Midnapore, 721 102, India

    Sukumar Mondal

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  1. Sambhu Charan Barman
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  2. Sukumar Mondal
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  3. Madhumangal Pal
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Correspondence to Sambhu Charan Barman.

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Barman, S.C., Mondal, S. & Pal, M. Minimum 2-tuple dominating set of permutation graphs. J. Appl. Math. Comput. 43, 133–150 (2013). https://doi.org/10.1007/s12190-013-0656-2

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  • DOI: https://doi.org/10.1007/s12190-013-0656-2

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