An improved upper bound for the domination number of a graph

Abstract

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor \frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma (G)\le \lfloor \sqrt{n}\rfloor$ which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph $G$ to have $\gamma (G)\le \lfloor \sqrt{n}\rfloor$, and some conditions sufficient for a graph $G$ to have $\gamma (G)\le \lfloor \sqrt{n}\rfloor$. We also present a characterization of all connected graphs $G$ of order $n$ with $\gamma (G)=\lfloor \sqrt{n}\rfloor$. Further, we prove that for a graph $G$ not satisfying $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$, deciding whether $\gamma (G)\le \lfloor \sqrt{n}\rfloor$ or $\gamma (\overline{G})\le \lfloor \sqrt{n}\rfloor$ can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph $G$.