A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function f is the sum, u∈V (G) f (u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.2010 Mathematics Subject Classification. 05C69.

For two vertices u and v of a graph G, the set I[u, v] consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v ∈ S. A set of vertices S ⊆ V(G) is a total geodetic set if I[S] = V(G) and the subgraph G[S] induced by S has no isolated vertex. The total geodetic number, denoted by t (G), is the minimum cardinality among all total geodetic sets of G. In this paper, we characterize all connected graphs G of order n ≥ 3 with t (G) = n − 1.