Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$for some constant c independent of d. This result is sharp in the sense that as $d \rightarrow \infty$ , almost all d-regular n-vertex graphs G of girth at least five have$$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result.

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五围图中的独立支配集

让$\伽马(G)$和$${\gamma _ \circ }(G)$$分别表示图 G 中最小支配集和最小独立支配集的大小。概率组合学的第一个结果是，如果G是一个n- 至少最小度数的顶点图d， 然后$$\begin{方程}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{方程}$$本文的主要结果是，如果G是任何n-顶点d- 至少五个周长的正则图，然后$$\begin{方程}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{方程}$$对于一些常数C独立于d. 这个结果在某种意义上是尖锐的，因为$d \rightarrow \infty$， 几乎全部d-常规的n- 至少五个周长的顶点图 G$$\begin{方程}\gamma_(G) \sim \frac{n}{d}\log d.\end{方程}$$此外，如果G是一个不相交的联合${n}/{(2d)}$完全二部图$K_{d,d}$， 然后${\gamma_\circ}(G) = \frac{n}{2}$. 我们也证明有n- 最小度数的顶点图 Gd并且其最大度数的增长速度不会比d日志d这样${\gamma_\circ}(G) \sim {n}/{2}$作为$d \rightarrow \infty$. 因此，主要结果需要周长条件和规则条件。