Graphs with Large Vertex–Edge Roman {2}-Domination Number

For a graph G=(V,E), a function f:V⟶{0,1,2} is a vertex–edge Roman {2}-dominating function (veR2DF) if for every edge xy∈E, either f(x)+f(y)≥1 or ∑z∈N(x)∪N(y)f(z)≥2. The sum ∑v∈Vf(v) is the weight of the function f. The vertex–edge Roman {2}-domination number of G, denoted by γveR2(G), is the minimum weight of a veR2DF of G. In this paper, we prove that for any connected graph G of order n≥4, then γveR2(G)≤n2, and this result is sharp. We also prove Nordhaus–Gaddum inequalities for γveR2(G). In addition, we characterize trees with equal vertex–edge Roman {2}-domination, vertex–edge domination, and domination numbers.