Total Roman domination in trees

A total Roman dominating function (TRDF) on a graph G is a function f:V(G) →{۰,۱,۲} satisfying the following conditions: (i) every vertex u for which f(u)=۰ is adjacent to at least one vertex v for which f(v)=۲ and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value, f(V(G))=Σu ∈V(G)f(u) . The total Roman domination number is the minimum weight of a total Roman dominating function of G. In this paper, we provide a lower bound on the total Roman domination number of a tree T in terms of its order and the number of leaves.

Keywords:

Total domination , total Roman domination number , trees