Signed total Roman domination in graphs

Let (G) be a finite and simple graph with vertex set (V(G)). A signed total Roman dominating function (STRDF) on a graph (G) is a function (f:V(G)rightarrow {-1,1,2}) satisfying the conditions that (i) (sum _{xin N(v)}f(x)ge 1) for each vertex (vin V(G)), where (N(v)) is the neighborhood of (v), and (ii) every vertex (u) for which (f(u)=-1) is adjacent to at least one vertex (v) for which (f(v)=2). The weight of an SRTDF (f) is (sum _{vin V(G)}f(v)). The signed total Roman domination number (gamma _{stR}(G)) of (G) is the minimum weight of an STRDF on (G). In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on (gamma _{stR}(G)). In addition, we determine the signed total Roman domination number of some classes of graphs.