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∈Nvf (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.