Signed total Roman domination in graphs

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