More results on the signed double Roman k-domination in graphs

Let k≥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)→{-1,1,2,3} satisfying the conditions that ∑x∈N[v]f(x)≥k for each vertex v∈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)≥2. The weight of an SDRkDF f is w(f)=∑v∈V(G)f(v). The signed double Roman k-domination number γsdRk(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 γsdRk(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, γsdR(G)=γsdR1(G), introduced and investigated in [1, 2].