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) ? 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 γ s R k ( 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 γ s R 2 ( T ) ? 10 n + 24 17 and we characterize the infinite family of trees that achieve equality in this bound.