further results on generous roman domination

Let g = (v (g), e(g)) be a graph and h be a function defined from v (g) to {0, 1, 2, 3}. a vertex x with h(x) = 0 is said to be undefended with respect to h if it has no neighbor assigned 2 or 3 under h. the function h is called a generous roman dominating function (grd-function) if for every vertex with h(x) = 0 there exists at least a vertex y with h(y) ≥ 2 adjacent to x such that the function η : v (g) → {0, 1, 2, 3}, defined by η(x) = α, η(y) = h(y) − α, where α ∈ {1, 2}, and η(z) = h(z) if z ∈ v (g) − {x, y} has no undefended vertex. the weight of a grd-function h is the value ∑ x∈v (g) h(x), and the minimum weight of a grd-function on g is the generous roman domination number (grd-number) of g. in this paper, we determine the exact value of the grd-number for the ladder graphs, and we provide an upper bound on it for trees in terms of the order, the number of leaves and the number of stems. moreover, we show that for every tree on at least three vertices, the grd-number is bounded below by the domination number plus 2, and we characterize the extremal trees attaining this lower bound.