Signed Roman k-domination in trees

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) Sigma(x is an element of N[v]) f(x) >= k for each vertex v is an element of 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 Sigma(v is an element of V(G)) f(v). The signed Roman k-domination number gamma(k)(sR)(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 gamma(2)(sR) (T) >= 10n+24/17 and we characterize the infinite family of trees that achieve equality in this bound. (C) 2015 Elsevier B.V. All rights reserved.