Abstract :
Let G = (V, E) be a simple graph. A set M ⊆ E is a matching if no two edges in M have a common vertex. The matching number, denoted β(G) (or β), is the maximum size of a matching in G. A double Roman dominating function (DRDF) on a graph G is a function f : V→ {0, 1, 2, 3} satisfying the conditions that for every vertex u of weight f (u) ∈ {0, 1}: (i ) if f (u) = 0, then u is adjacent to either at least one vertex v with f (v) = 3 or two vertices v1, v2 with f (v1) = f (v2) = 2. (ii ) if f (u) = 1, then u is adjacent to at least one vertex v with f (v) ∈ {2, 3}. The weight of a double Roman dominating function f is the value f (V) = u∈V f (u). The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number of G, denoted by γdR (G). In this paper, first, we note that γdR(G) ≤ 3β(G), where G is a graph without isolated vertices. Moreover, we give a descriptive characterization of block graphs G satisfying γdR(G) = 3β(G). Finally, we show that the decision problem associated with γdR(G) = 3β(G) is CO −NP-complete for bipartite graphs.
