Upper bounds for covering total double Roman domination

Abstract

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 ?ctdR(G) is the minimum weight of a CT DRD function on G. In this work, we present some contributions to the study of ?ctdR(G)-function of graphs. For the non star trees T, we show that ?ctdR(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, ?ctdR(G) ? 4n/3 and finally, we show that, for a simple graph G of order n with ?(G) ? 2, ?ctdR(G) ? 4n/3 and this bound is sharp.