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 $$\gamma _{stR}(G)$$ γ s t R ( 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 $$\gamma _{stR}(G)$$ γ s t R ( G ) . In addition, we determine the signed total Roman domination number of some classes of graphs.