Double Roman and double Italian domination

Abstract

Let G be a graph with vertex set V (G). A double Roman dominating function (DRDF) on a graph G is a function f : V (G) −→ {0, 1, 2, 3} that satisfies the following conditions: (i) If f (v) = 0, then v must have a neighbor w with f (w) = 3 or two neighbors x and y with f (x) = f (y) = 2; (ii) If f (v) = 1, then v must have a neighbor w with f (w) ≥ 2. The weight of a DRDF f is the sum v∈V (G) f (v). The double Roman domination number equals the minimum weight of a double Roman dominating function on G. A double Italian dominating function (DIDF) is a function f : V (G) −→ {0, 1, 2, 3} having the property that f (N [u]) ≥ 3 for every vertex u ∈ V (G) with f (u) ∈ {0, 1}, where N [u] is the closed neighborhood of v. The weight of a DIDF f is the sum v∈V (G) f (v), and the minimum weight of a DIDF in a graph G is the double Italian domination number. In this paper we first present Nordhaus-Gaddum type bounds on the double Roman domination number which improved corresponding results given in [N. Jafari Rad and H. Rahbani, Some progress on the double Roman domination in graphs, Discuss. Math. Graph Theory 39 (2019) 41-53]. Furthermore, we establish lower bounds on the double Roman and double Italian domination numbers of trees.