Let \(k\ge 1\) be an integer, and let G be a finite and simple graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is defined in [Signed double Roman k-domination in graphs, Australas. J. Combin. 72 (2018), 82–105] as a function \(f :V(G) \rightarrow \{-1,1,2,3\}\) satisfying the conditions that \(\sum _{x\in N[v]}f(x)\ge k\) for each vertex \(v\in V(G)\) , where N[v] is the closed neighborhood of v, every vertex u for which \(f(u)=-1\) is adjacent to at least one vertex v for which \(f(v)=3\) or adjacent to two vertices x and y with \(f(x)=f(y)=2\) , and every vertex u with \(f(u)=1\) is adjacent to vertex v with \(f(v)\ge 2\) . The weight of an SDRkDF f is \(\textrm{w}(f) = \sum _{v\in V(G)}f(v)\) . The signed double Roman k-domination number \(\gamma _{\textrm{sdR}}^k(G)\) of G is the minimum weight among all SDRkDF on G. In this paper we continue the study of the signed double Roman k-domination number of graphs, and we present new bounds on \(\gamma _{\textrm{sdR}}^k(G)\) . In addition, we determine the signed double Roman k-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed double Roman domination number, \(\gamma _{\textrm{sdR}}(G)=\gamma _{\textrm{sdR}}^1(G)\) , introduced and investigated in [1, 2].
