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An Algorithm for the Inverse Distance-2 Dominating Set of a Graph

  • K. Ameenal Bibi
  • A. Lakshmi
  • R. Jothilakshmi
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Let G = (V, E) be a simple, finite, connected, and undirected graph. Let D ⊆ V (G) be the non-empty subset of V (G) such that D is the minimum distance-2 dominating set in the graph G = (V, E). If V − D contains a distance-2 dominating set D of G, then D is called an inverse distance-2 dominating set with respect to D. The inverse distance-2 domination number \({{\gamma }_{\leq 2}}^{-1}\left (G\right )\) of G is the minimum cardinality of the minimal inverse distance-2 dominating set of G. In this paper, we presented an algorithm for finding an inverse distance-2 dominating set of a graph.

References

  1. 1.
    K. Ameenal Bibi, A. Lakshmi and R. Jothilakshmi, (2017), An Algorithm for Minimal and Minimum distance - 2 dominating sets of Graph, Global Journal of pure and applied Mathematics, ISSN -0978-1768 vol.13, No.2, pp. 1117–1126.Google Scholar
  2. 2.
    E.J. Cockayne, S.T. Hedetniemi, (1977) “Towards a Theory of Domination in Graphs”, Networks, 7, pp. 247–261.MathSciNetCrossRefGoogle Scholar
  3. 3.
    T.H. Cormen, C.E. Leiserson, R.L. Riest and C. Stein, “Introduction to Algorithms”, 2nd Edition, MIT Press, 2001.Google Scholar
  4. 4.
    B. Das and V. Bharghavan , “Routing in Ad Hoc Networks Using Minimum Connected Dominating set”, in Proceedings of International Conference on Communications’ 97, Montreal, Canada. June 1997.Google Scholar
  5. 5.
    J.R Griggs and J.P. Hutchinson, “On the r-domination number of a graph”, Discrete Mathematics, 101, 1992, pp. 65–72.MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Guha and S. Khuller, ”Approximation Algorithm for connected dominating sets”, Algorithmica, vol. 20, no.4, pp. 374–387. Apr. 1998.Google Scholar
  7. 7.
    F. Harary, “Graph Theory”, Addison – wesley, Massachusetts, 1969.CrossRefGoogle Scholar
  8. 8.
    T.W. Haynes, S.T. Hedetniemi, P.J. Slater, “Fundamentals of Domination in Graphs”, Marcel Dekker Publishers, New York, 1998.zbMATHGoogle Scholar
  9. 9.
    Mallikarjun Avula, Seong-Moo and Seungjin Park, “Constructing Minimum Connected Dominating Set in Mobile Ad Hoc Networks”, published in International journal on applications of graph theory in wireless ad hoc networks and sensor networks, Vol 4, No.2/3, Sep 2012, pp. 15–27.Google Scholar
  10. 10.
    Mano Yadav, Vinay Rishiwal, G. Arora and S. Makka, “Modified Minimum Connected Dominating set formation for Wireless Adhoc Networks”, in Journal of Computing, Vol 1, Issue 1, Dec 2009, pp. 200–203.Google Scholar
  11. 11.
    K. Sakai, M.T. Sun and W.S. Ku and Hiromi Okada, “Maintaining CDS in Mobile Ad hoc Networks”, wireless Algorithms Systems and Applications, Lecture Note in Computer Science, 2008, Vol 5258, pp. 141–153.CrossRefGoogle Scholar
  12. 12.
    P.J. Slater,”R-Domination in Graphs,” Journal of Association for Computer Machinery, 23(3), July 1976, pp. 446–450.CrossRefGoogle Scholar
  13. 13.
    N. Sridharan, V.S.A. Subramanian and M.D. Elias, “Bounds on the Distance Two- Domination Number of a Graph, Graphs and Combinatorics”, Journal of Association for Computer Machinery,18(3), 2002, pp. 997–675.Google Scholar
  14. 14.
    S.K. Vaidya, N.J. Kothari, “Distance k-domination of some path related graphs”, International Journal of Mathematics and Soft Computing, Vol. 4, 2014, pp. 1–5.CrossRefGoogle Scholar
  15. 15.
    D.B. West, Introduction to Graph Theory, 2nd Ed., Prentice Hall, Upper Saddle River, NJ, 2001, pp.116–118.Google Scholar
  16. 16.
    J. Wu and H.L. Li, “On Calculating Connected Dominating Set for Efficient Routing in Ad Hoc Wireless Networks”, in Proceedings of the 3rd ACM International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, 1999, pp 7–14.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • K. Ameenal Bibi
    • 1
  • A. Lakshmi
    • 1
  • R. Jothilakshmi
    • 2
  1. 1.PG and Research Department of MathematicsD.K.M College for Women (Autonomous)VelloreIndia
  2. 2.PG and Research Department of MathematicsMazharul Uloom CollegeAmburIndia

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