Note on the total ۲-rainbow domination number in trees

A ۲-rainbow dominating function (۲RDF) of a graph 𝐺 = (𝑉 (𝐺),𝐸(𝐺)) is a function 𝑓 from the vertex set 𝑉(𝐺) to the set of all subsets of the set {۱, ۲} such that for every vertex v ∈ V (G) with f(v) = ∅ the condition ⋃u∈N(v) f(u) = {۱, ۲} is fulfilled, where 𝑁(𝑣) is the open neighborhood of 𝑣. A total ۲-rainbow dominating function 𝑓 of a graph with no isolated vertices is a ۲RDF with the additional condition that the subgraph of G induced by {v ∈ V (G) | f(v) ≠∅} has no isolated vertex. The total ۲-rainbow domination number, 𝛾𝑡𝑟۲(G), is the minimum weight of a total ۲-rainbow dominating function of G. In this paper, we provide a lower bound on the total ۲-rainbow domination number of a tree 𝑇 in terms of its order, the number of support vertices and leaves

Keywords:

Total domination , total ۲-rainbow domination number , tree.