Real Domination in Graphs

A function \(f: V \rightarrow \mathbb{R}\) is defined to be an \(\mathbb{R}\)-dominating function of graph \(G = (V, E)\) if the sum of the function values over any closed neighbourhood is at least 1. That is, for every \(v \in V\),
\(f(N(v) \cup \{v\}) \geq 1\).

The \(\mathbb{R}\)-domination number \(\gamma_{\mathbb{R}}(G)\) of \(G\) is defined to be the infimum of \(f(V)\) taken over all \(\mathbb{R}\)-dominating functions \(f\) of \(G\).

In this paper, we investigate necessary and sufficient conditions for \(\gamma_{\mathbb{R}}(G) = \gamma(G)\), where \(\gamma(G)\) is the standard domination number.