Upper bounds for covering total double Roman domination

Let G = (V, E) be a finite simple graph where V = V (G) and E = E(G). Suppose that G has no isolated vertex. A covering total double Roman dominating function (CT DRD function) f of G is a total double Roman dominating function (T DRD function) of G for which the set {v ∈ V (G)|f(v) ≠ 0} is a covering set. The covering total double Roman domination number γctdR(G) is the minimum weight of a CT DRD function on G. In this work, we present some contributions to the study of γctdR(G)-function of graphs. For the non star trees T, we show that γctdR(T) ≤ 4n(T )+5s(T )−4l(T )/3, where n(T), s(T) and l(T) are the order, the number of support vertices and the number of leaves of T respectively. Moreover, we characterize trees T achieve this bound. Then we study the upper bound of the 2-edge connected graphs and show that, for a 2-edge connected graphs G, γctdR(G) ≤ 4n/3 and finally, we show that, for a simple graph G of order n with δ(G) ≥ 2, γctdR(G) ≤ 4n/3 and this bound is sharp.Publisher's Versio

Some Progress on the Double Roman Domination in Graphs

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 ω with f(ω) ≥ 2. The weight of a DRDF f is the sum f(V ) = Σv∈Vf(v), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γdR(G). In this paper, we derive sharp upper and lower bounds on γdR(G) + γdR(Ḡ) and also γdR(G)γdR(Ḡ) ,where Ḡ is the complement of graph G. We also show that the decision problem for the double Roman domination number is NP- complete even when restricted to bipartite graphs and chordal graphs