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 $G$ 为阶数为 $n$ 的图。对于无孤立顶点的图 $G$，其支配数的一个经典上界为 $\lfloor\frac{n}{2}\rfloor$。然而，对于若干图族，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 $\gamma(G) \le \lfloor\sqrt{n}\rfloor$ 的必要条件及若干充分条件。我们还对所有满足 $\gamma(G) = \lfloor\sqrt{n}\rfloor$ 的 $n$ 阶连通图 $G$ 进行了刻画。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 $G$，判定 $\gamma(G) \le \lfloor\sqrt{n}\rfloor$ 或 $\gamma(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ 是否成立可在多项式时间内完成。我们猜想该判定问题对任意图 $G$ 均可在多项式时间内求解。