Roman domination number of Generalized Petersen Graphs P(n, 2)

A Roman domination function on a graph G = (V,E) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman domination function f is the value f(V (G)) = ∑ u∈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 this paper, we study the Roman domination number of generalized Petersen graphs P (n, 2) and prove that γR(P (n, 2)) = ⌈ 8n 7 ⌉(n ≥ 5).