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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

Abstract

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.

Keywords

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

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