Geodesic
Extremal Graphs for a Bound on the Roman Domination Number
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 3, pp. 771-785

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A Roman dominating function on a graph G = (V, E) is a function f:V (G) → 0, 1, 2 such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value w(f) = Σu∈V(G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by γR(G). In 2009, Chambers, Kinnersley, Prince and West proved that for any graph G with n vertices and maximum degree Δ, γR(G) ≤ n + 1 − Δ. In this paper, we give a characterization of graphs attaining the previous bound including trees, regular and semiregular graphs. Moreover, we prove that the problem of deciding whether γR(G) = n + 1 − Δ is co-complete. Finally, we provide a characterization of extremal graphs of a Nordhaus–Gaddum bound for γR(G) + γR (Ḡ), where Ḡ is the complement graph of G.
Keywords: Roman domination, Roman domination number, Nordhaus-Gaddum inequalities 
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Bouchou, Ahmed; Blidia, Mostafa; Chellali, Mustapha. Extremal Graphs for a Bound on the Roman Domination Number. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 3, pp. 771-785. http://geodesic.mathdoc.fr/item/DMGT_2020_40_3_a4/