total restrained roman domination

Let g be a graph with vertex set v (g). a roman dominating function (rdf) on a graph g is a function f : v (g) −→ {0, 1, 2} such that every vertex v with f(v) = 0 is adjacent to a vertex u with f(u) = 2. if f is an rdf on g, then let vi = {v ∈ v (g) : f(v) = i} for i ∈ {0, 1, 2}. an rdf f is called a restrained (total) roman dominating function if the subgraph induced by v0 (induced by v1 ∪v2) has no isolated vertex. a total and restrained roman dominating function is a total restrained roman dominating function. the total restrained roman domination number γtrr(g) on a graph g is the minimum weight of a total restrained roman dominating function on the graph g. we initiate the study of total restrained roman domination number and present several sharp bounds on γtrr(g). in addition, we determine this parameter for some classes of graphs.