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Abstract
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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.
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