مشخصات پژوهش

A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. A set {f1, f2, . . . , fd} of total Roman dominating functions on G with the property that Pd i=1 fi(v) ≤ 2 for each v ∈ V (G), is called a total Roman dominating family (of functions) on G. The maximum number of functions in a total Roman dominating family on G is the total Roman domatic number of G, denoted by dtR(G). In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for dtR(G) and we determine the total Roman domatic number of some special graphs.