Lower Bounds on the Signed \(k\)-Domination Number of Graphs

Abstract

Let \(G\) be a graph with vertex set \(V(G)\). For any integer \(k \geq 1\), a signed \(k\)-dominating function is a function \(f: V(G) \rightarrow \{-1, 1\}\) satisfying \(\sum_{x \in N[v]} f(t) \geq k\) for every \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v\). The minimum of the values \(\sum_{v \in V(G)} f(v)\), taken over all signed \(k\)-dominating functions \(f\), is called the signed \(k\)-domination number. In this note, we present some new lower bounds on the signed \(k\)-domination number of a graph. Some of our results improve known bounds.