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.