Equitable Power Domination Number of Certain Graphs

  • Authors

    • S. Banu Priya
    • N. Srinivasan
    2018-10-02
    https://doi.org/10.14419/ijet.v7i4.10.20933
  • Dominating set, Equitable dominating set, Power dominating set, Equitable power dominating set, Equitable power domination number.

  • Abstract

    Let   12G">  be a graph with vertex set 12V"> , a set 12Sâٹ†V">  is said to be a power dominating set (PDS), if every vertex  12u∈V-S">  is observed by some vertices in 12S">  using the following rules: (i) if a vertex  12v">  in 12G">  is in PDS, then it dominates itself and all the adjacent vertices of 12v">  and (ii) if an observed vertex 12v">  in 12G">  has  12k>1">  adjacent vertices and if 12k-1">  of these vertices are already observed, then the remaining one non-observed vertex will also be observed by 12v">  in 12G"> . The degree 12d(v)">  of a vertex 12v">  in 12G">  is the number of edges of 12G">  incident with 12v">  and any two adjacent vertices 12u">  and 12v">  in 12G">  are said to hold equitable property if 12|d(u)-d(v)| ≤ 1"> . In this paper, we introduce the notions of equitable power dominating set and equitable power domination number. We also derive the equitable power domination number of certain graphs.

     

     

  • References

    1. [1] Hamada T & Yoshimura I (1976), Tra versibility and connectivity of the middle graph of a graph, Discrete Math., Vol.14, pp. 247-256.

      [2] Bermond JC (1979), Graceful Graphs, radio antennae and french windmills, Graph Theory and Combinatorics, pp. 18-37.

      [3] Bondy JA & Murthy USR (1986), Graph Theory withApplications, Elsevier, North Holland, New York.

      [4] Baldwin TL, Mili L, Boisen MB Jr & Adapa R(1993), Power system observability with minimal phasor measurement placement, IEEE Trans, Vol. 8, pp. 707- 715.

      [5] Barrera R & Ferrero D (2011), Power domination in cylinders and generalized petersen graphs, Networks, Vol. 58, pp. 43-49.

      [6] Brueni DJ & Heath LS (2005), The PMU placement problem, SIAM J. Dis crete Math, Vol. 19, No.3, pp. 744-761.

      [7] Gera R, Horton S & Rasmussen C (2006), Dominator colorings and safe clique partitions, Congressus Numerantium, Vol. 181, pp. 19-32.

      [8] Hedetniemi ST & Laskar RC (1990), Bibliography on domination in graphs and some basic definitions of domination parameters, https://en.wikipedia.org/wiki/Discrete_Mathematics_(journal)">Discrete Mathematics, Vol. 86, No. 1–3, pp. 257–277.

      [9] Hosoya H & Harary F (1993), On the matching properties of three fence graphs, J. Math. Chem. Vol. 12, pp. 211- 218.

      [10] Narasingh Deo (1994), Graph theory with applications to engineering and computer science, New Delhi.

      [11] Swaminathan V & Dharmalingam KM (2011), Degree equitable domination on graphs, Kragujevac Journal of Mathematics, Vol. 35, No. 1, pp. 191–197.

      [12] Anitha A, Arumugam S & Mustapha chellali (2011), Equitable domination in graphs, Discrete Mathematics, Algorithms and Applications, Vol. 03, pp. 311.

      [13] Agasthi P, Parvathi N & Thirusangu K (2017), On Some labelings of line graph of barbell graph, International Journal of Pure and Applied Mathematics, Vol. 113, No. 10, pp. 148-156.

      [14] Chen, Yichao, Gross, Jonathan L Mansour & Toufik (2013), Total embedding distributions of circular ladders, Journal of Graph Theory, Vol. 74(1), pp. 32-57.

  • Downloads

  • How to Cite

    Banu Priya, S., & Srinivasan, N. (2018). Equitable Power Domination Number of Certain Graphs. International Journal of Engineering & Technology, 7(4.10), 349-354. https://doi.org/10.14419/ijet.v7i4.10.20933

    Received date: 2018-10-04

    Accepted date: 2018-10-04

    Published date: 2018-10-02