Domination in Regular Graphs

A two-valued function \(f\) defined on the vertices of a graph \(G = (V,E)\), \(f: V \to \{-1,1\}\), is a signed dominating function if the sum of its function values over any closed neighborhoods is at least one. That is, for every \(v \in V\), \(f(N[v]) \geq 1\), where \(N[v]\) consists of \(v\) and every vertex adjacent to \(v\). The function \(f\) is a majority dominating function if for at least half the vertices \(v \in V\), \(f(N[v]) \geq 1\). The weight of a signed (majority) dominating function is \(f(V) = \sum f(v)\). The signed (majority) domination number of a graph \(G\), denoted \(\gamma_s(G)\) (\(\gamma_{\text{maj}}(G)\), respectively), equals the minimum weight of a signed (majority, respectively) dominating function of \(G\). In this paper, we establish an upper bound on \(\gamma_s(G)\) and a lower bound on \(\gamma_{\text{maj}}(G)\) for regular graphs \(G\).