Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON THE DOMINATION NUMBER OF A GRAPH AND ITS SQUARE GRAPH

  • Murugan, E. (Department of Mathematics, Manonmaniam Sundaranar University) ;
  • Joseph, J. Paulraj (Department of Mathematics, Manonmaniam Sundaranar University)
  • Received : 2021.05.08
  • Accepted : 2022.02.20
  • Published : 2022.06.30
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Abstract

For a given graph G = (V, E), a dominating set is a subset V' of the vertex set V so that each vertex in V ＼ V' is adjacent to a vertex in V'. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G). For an integer k ≥ 1, the k-th power Gk of a graph G with V (Gk) = V (G) for which uv ∈ E(Gk) if and only if 1 ≤ dG(u, v) ≤ k. Note that G2 is the square graph of a graph G. In this paper, we obtain some tight bounds for the sum of the domination numbers of a graph and its square graph in terms of the order, order and size, and maximum degree of the graph G. Also, we characterize such extremal graphs.

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References

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