Signed Roman [formula omitted]-domination in trees

Abstract

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