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 (CTDRD function) f of G is a total double Roman dominating function (TDRD function) of G for which the set {v [member of] V(G)|f(v) [not equal to] 0} is a covering set. The covering total double Roman domination number [[gamma].sub.ctdR](G) is the minimum weight of a CTDRD function on G. In this work, we present some contributions to the study of [[gamma].sub.ctdR](G)-function of graphs. For the non star trees T, we show that [Please download the PDF to view the mathematical expression] 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, [[gamma].sub.ctdR](G) [less than or equal to] [[4n]/[3]] and finally, we show that, for a simple graph G of order n with [delta](G) [greater than or equal to] 2, [[gamma].sub.ctdR](G) [less than or equal to] [[4n]/[3]] and this bound is sharp. Keywords: Total double Roman domination, covering, tree, upper bound. AMS Subject Classification: 05C69.