Equality in a bound that relates the size and the restrained domination number of a graph

Abstract

Let $G=(V,E)$ be a graph. A set $S\subseteq V$ is a restrained dominating set if every vertex in $V-S$ is adjacent to a vertex in $S$ and to a vertex in $V-S$. The restrained domination number of $G$, denoted $\gamma _{r}(G)$, is the smallest cardinality of a restrained dominating set of $G$. The best possible upper bound $q(n,k)$ is established in Joubert (Discrete Appl Math 161:829–837, 2013) on the size $m(G)$ of a graph $G$ with a given order $n \ge 5$ and restrained domination number $k \in \{3, \ldots , n-2\}$. We extend this result to include the cases $k=1,2,n$, and characterize graphs $G$ of order $n \ge 1$ and restrained domination number $k \in \{1,\dots , n-2,n\}$ for which $m(G)=q(n,k)$.