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 V-i = {v is an element of V (G) : f(v) = i} for i is an element of {0; 1; 2}. An RDF f is called a restrained (total) Roman dominating function if the subgraph induced by V0 (induced by V-1 boolean OR V-2) has no isolated vertex. A total and restrained Roman dominating function is a total restrained Roman dominating function. The total restrained Roman domination number gamma(trR)(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 gamma(trR)(G). In addition, we determine this parameter for some classes of graphs.