Some Cycle and Star Related Nordhaus-Gaddum Type Relations on Strong Efficient Dominating Sets

  • Murugan, Karthikeyan (Department of Mathematics, The M. D. T. Hindu College and Manonmaniam Sundaranar University)
  • Received : 2016.11.22
  • Accepted : 2018.06.08
  • Published : 2019.09.23


Let G = (V, E) be a simple graph with p vertices and q edges. A subset S of V (G) is called a strong (weak) efficient dominating set of G if for every $v{\in}V(G)$ we have ${\mid}N_s[v]{\cap}S{\mid}=1$ (resp. ${\mid}N_w[v]{\cap}S{\mid}=1$), where $N_s(v)=\{u{\in}V(G):uv{\in}E(G),\;deg(u){\geq}deg(v)\}$. The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak) efficient domination number of G and is denoted by ${\gamma}_{se}(G)$ (${\gamma}_{we}(G)$). A graph G is strong efficient if there exists a strong efficient dominating set of G. In this paper, some cycle and star related Nordhaus-Gaddum type relations on strong efficient dominating sets and the number of strong efficient dominating sets are studied.


strong efficient dominating sets;strong efficient domination number and number of strong efficient dominating sets


  1. D. W. Bange, A. E. Barkauskas and P. J. Slater, Efficient dominating sets in graphs, Application of Discrete Mathematics, SIAM, Philadephia, (1988), 189-199.
  2. G. Chartrand, H. Hevia, E. B. Jarette and M. Schultz, Subgraph distances in graphs defined by edge transfers, Discrete Math., 170(1997), 63-79.
  3. F. Harary, Graph theory, Addison-Wesley, 1969.
  4. F. Harary, T. W. Haynes and P. J. Slater, Efficient and excess domination in graphs, J. Combin. Math. Combin. Comput., 26(1998), 83-95.
  5. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc, New York, 1998.
  6. T. W. Haynes, M. A. Henning, P. J. Slater and L. C. Van Der Merwe, The complementary product of two graphs, Bull. Inst. Combin. Appl., 51(2007), 21-30.
  7. N. Meena, Studies in graph theory-efficient domination and related topics, Ph. D. Thesis, Manonmaniam Sundaranar University, 2013.
  8. K. Murugan and N. Meena, Some Nordhaus-Gaddum type relation on strong efficient dominating sets, J. New Results Sci., 5(11)(2016), 4-16.
  9. E. Sampathkumar and S. B. Chikkodimath, Semi-total graphs of a graph I, J. Kar-natak Univ. Sci., 18(1973), 274-280.
  10. E. Sampathkumar and L. P. Latha, Strong weak domination and domination balance in a graph, Discrete Math., 161(1996), 235-242.
  11. D. V. S. S. Sastry and B.S. P. Raju, Graph equations for line graphs, total graphs, middle graphs and quasitotal graphs, Discrete Math., 48(1984), 113-119.
  12. H. Whitney, Congruent graphs and the connectivity graphs, Amer. J. Math., 54(1932), 150-168.