On Double Roman Dominating Functions in Graphs

Abstract

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is a double Roman dominating function of G if for each v ∈ V (G) with f(v) = 0, v has two adjacent vertices u and w for which f(u) = f(w) = 2 or v has an adjacent vertex u for which f(u) = 3, and for each v ∈ V (G) with f(v) = 1, v is adjacent to a vertex u for which either f(u) = 2 or f(u) = 3. The minimum weight ωG(f) = P v∈V (G) f(v) of a double Roman dominating function f of G is the double Roman domination number of G. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of graphs.