On the Global Distance Roman Domination of Some Graphs

Abstract

Let k ∈ Z +. A k − distance Roman dominating function (kDRDF) on G = (V, E) is a function f : V → {0, 1, 2} such that for every vertex v with f(v) = 0, there is a vertex u with f(u) = 2 with d(u, v) ≤ k. The function f is a global k − distance Roman dominating function (GkDRDF) on G if and only if f is a k − distance Roman dominating function (kDRDF) on G and on its complement G. The weight of the global k − distance Roman dominating function (GkDRDF) f is the value w(f) = P x∈V f(x). The minimum weight of the global k − distance Roman dominating function (GkDRDF) on the graph G is called the global k − distance Roman domination number of G and is denoted as γ k gR(G). A γ k gR(G) − function is the global k − distance Roman dominating function on G with weight γ k gR(G). Note that, the global 1 − distance Roman domination number γ 1 gR(G) is the usual global Roman domination number γgR(G), that is, γ 1 gR(G) = γgR(G). The authors initiated this study. In this paper, the authors obtained and established the following results: preliminary results on global distance Roman domination; the global distance Roman domination on Kn, Kn, Pn, and Cn; and, some bounds and characterizations of global distance Roman domination over any graphs.