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 $G$.

翻译：设 $G$ 是一个 $n$ 阶图。经典结论指出，不含孤立点的图 $G$ 的控制数上界为 $\lfloor\frac{n}{2}\rfloor$。然而，对于若干图族，我们有 $\gamma(G) \le \lfloor\sqrt{n}\rfloor$，这给出了一个显著改进的上界。本文给出了图 $G$ 满足 $\gamma(G) \le \lfloor\sqrt{n}\rfloor$ 的必要条件，以及若干充分条件。此外，我们刻画了所有满足 $\gamma(G) = \lfloor\sqrt{n}\rfloor$ 的 $n$ 阶连通图 $G$ 的特征。进一步，我们证明：对于不满足 $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$ 的图 $G$，判断是否满足 $\gamma(G) \le \lfloor\sqrt{n}\rfloor$ 或 $\gamma(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ 的问题可在多项式时间内解决。我们猜想该判定问题对于任意图 $G$ 均可在多项式时间内解决。