On signed total double Romanmination number of graphs

A signed total double Roman dominating function (STDRD-function) on an isolated-free graph 𝐺 is a function 𝑓:𝑉⁡(𝐺)→{−1,1,2,3} satisfying the conditions (i) 𝑓⁡(𝑁⁡(𝑣))=∑𝑧∈𝑁⁡(𝑣)‍𝑓⁡(𝑧)≥1 for every vertex 𝑣∈𝑉⁡(𝐺) and (ii) if 𝑓⁡(𝑣)=−1, then the vertex 𝑣 must have a neighbor assigned 3 or two neighbors assigned 2 under 𝑓, and if 𝑓⁡(𝑣)=1, then 𝑣 must have at least one neighbor assigned at least 2. The weight of an STDRD-function 𝑓 is the value 𝑓⁡(𝑉⁡(𝐺))=∑𝑥∈𝑉⁡(𝐺)‍𝑓⁡(𝑥), and the signed total double Roman domination number (or simply STDRD-number) 𝛾𝑡𝑠𝑑𝑅⁡(𝐺) of 𝐺 is the minimum weight of an STDRD-function of 𝐺. In this work, we establish several new bounds for the STDRD-number, which refine and extend previously known results. Moreover, we provide an exact determination of the STDRD-number in the case of perfect binary trees.