Real Domination in Graphs

Abstract

A function $f:V\to \mathbb{R}$ 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 $v\in V$,
$f(N(v)\cup \{v\})\ge 1$.

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.