On the differential and Roman domination number of a graph with minimum degree two

Let G=(V,E) be a graph of order n and let B(D) be the set of vertices in V∖D that have a neighbor in the vertex set D. The differential of a vertex set D is defined as ∂(D)=|B(D)|−|D| and the maximum value of ∂(D) for any subset D of V is the differential of G, denoted by ∂(G). A Roman dominating function of G is a function f:V→{0,1,2} such that every vertex u with f(u)=0 is adjacent to a vertex v with f(v)=2. The weight of a Roman dominating function is the value f(V)=∑u∈Vf(u). The minimum weight of a Roman dominating function of a graph G is the Roman domination number of G, written γR(G). Bermudo, et al. proved that these two parameters are complementary with respect to the order n of the graph, that is, ∂(G)+γR(G)=n. In this work we prove that, for any graph G with order n≥9 and minimum degree two, ∂(G)≥3γ(G)4, consequently, γR(G)≤n−3γ(G)4, where γ(G) is the domination number of G. We also prove that for any graph with order n≥15, minimum degree two and without any induced tailed 5-cycle graph of seven vertices or tailed 5-cycle graph of seven vertices together with a particular edge, it is satisfied ∂(G)≥5n17, consequently, γR(G)≤12n17.