Lower bounds for signed double Roman k-domination in unicyclic graphs

Let 𝑘≥1 be an integer and let 𝐺 denote a finite and simple graph with vertex set 𝑉⁡(𝐺). A signed double Roman 𝑘-dominating function on 𝐺 is a mapping 𝑓:𝑉⁡(𝐺)→{−1,1,2,3} satisfying the following: (i) if 𝑓⁡(𝑣)=−1 for a vertex 𝑣, then 𝑣 is adjacent either to a vertex 𝑤 with 𝑓⁡(𝑤)=3 or to at least two vertices assigned value 2 under 𝑓; (ii) if 𝑓⁡(𝑣)=1, then 𝑣 has a neighbor 𝑤 with 𝑓⁡(𝑤)≥2; and (iii) for every vertex 𝑣, ∑𝑢∈𝑁⁡[𝑣]𝑓⁡(𝑢)≥𝑘. The weight of a signed double Roman 𝑘-dominating function 𝑓 is given by ∑𝑢∈𝑉⁡(𝐺)𝑓⁡(𝑢), and the minimum possible weight is called the signed double Roman 𝑘-domination number of 𝐺. In this paper, we investigate the signed double Roman 𝑘-domination number in unicyclic graphs, where lower bounds are established for 𝑘∈{1,2,3,4}. Moreover, a characterization of extremal unicyclic graphs reaching these bounds when 𝑘∈{1,2} is provided.