مشخصات پژوهش

A double Roman dominating function (DRDF) on a graph G = (V;E) is a function f : V ! f0; 1; 2; 3g having the property that if f(u) = 0, then a vertex u has at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, and if f(u) = 1, then a vertex u must have at least one neighbor w with f(w)  2. A total double Roman dominating function (TDRDF) on a graph G with no isolated vertex is a DRDF f on G with the additional property that the subgraph of G induced by the set fv 2 V : f(v) 6= 0g has no isolated vertices. The weight of a total double Roman dominating function f is the value, f(V ) = u2V (G)f(u). The total double Roman domination number tdR(G) is the minimum weight of a TDRDF on G. A subset S of V is a 2-independent set of G if every vertex of S has at most one neighbor in S. The maximum cardinality of a 2-independent set of G is the 2-independence number 2(G). In this paper, we show that if T is a tree, then tdR(T)  22(T), and we characterize all trees attaining the equality.