For a graph G = (V, E), a double Roman dominating function (or just DRDF) is a function f : V −→ {0, 1, 2, 3} having the property that if f (v) = 0 for a vertex v, then v has at least two neighbors assigned 2 under f or one neighbor assigned 3 under f , and if f (v) = 1, then vertex v must have at least one neighbor w with f (w) ≥ 2. The weight of a DRDF f is the sum f (V) = v∈V f (v), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γ dR (G). In this paper, we derive sharp upper and lower bounds on γ dR (G) + γ dR (G) and also γ dR (G)γ dR (G), where G is the complement of graph G. We also show that the decision problem for the double Roman domination number is NPcomplete even when restricted to bipartite graphs and chordal graphs.
Let G = ( V , E ) be a graph and f : V ⟶ { 0 , 1 , 2 } be a function. Given a vertex u with f ( u ) = 0 , if all neighbors of u have zero weights, then u is called undefended with respect to f. Furthermore, if every vertex u with f ( u ) = 0 has a neighbor v with f ( v ) > 0 and the function f ′ : V ⟶ { 0 , 1 , 2 } with f ′ ( u ) = 1 , f ′ ( v ) = f ( v ) − 1 , f ′ ( w ) = f ( w ) if w ∈ V ∖ { u , v } has no undefended vertex, then f is called a weak Roman dominating function. Also, the function f is a perfect Roman dominating function if every vertex u with f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . Let the weight of f be w ( f ) = ∑ v ∈ V f ( v ) . The weak (resp., perfect) Roman domination number, denoted by γ r ( G ) (resp., γ R p ( G ) ), is the minimum weight of the weak (resp., perfect) Roman dominating function in G. In this paper, we characterize those trees where the perfect Roman domination number strongly equals the weak Roman domination number, in the sense that each weak Roman dominating function of minimum weight is, at the same time, perfect Roman dominating.
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