Signed Roman kk-domination in trees

Let k≥1k≥1 be an integer, and let GG be a finite and simple graph with vertex set V(G)V(G). A signed Roman kk-dominating function (SRkDF) on a graph GG is a function f:V(G)→{−1,1,2}f:V(G)→{−1,1,2} satisfying the conditions that (i) ∑x∈N[v]f(x)≥k∑x∈N[v]f(x)≥k for each vertex v∈V(D)v∈V(D), where N[v]N[v] is the closed neighborhood of vv, and (ii) every vertex uu for which f(u)=−1f(u)=−1 is adjacent to at least one vertex vv for which f(v)=2f(v)=2. The weight of an SRkDF ff is ∑v∈V(G)f(v)∑v∈V(G)f(v). The signed Roman kk-domination number γsRk(G) of GG is the minimum weight of an SRkDF on GG. In this paper we establish a tight lower bound on the signed Roman 22-domination number of a tree in terms of its order. We prove that if TT is a tree of order n≥4n≥4, then γsR2(T)≥10n+2417 and we characterize the infinite family of trees that achieve equality in this bound.