A permutation graph $G_\pi$ is a simple graph with vertices corresponding to the elements of $\pi$ and an edge between $i$ and $j$ when $i$ and $j$ are inverted in $\pi$. A set of vertices $D$ is said to dominate a graph $G$ when every vertex in $G$ is either an element of $D$, or adjacent to an element of $D$. The domination number $\gamma(G)$ is defined as the cardinality of a minimum dominating set of $G$. A strong fixed point of a permutation $\pi$ of order $n$ is an element $k$ such that $\pi^{-1}(j) \pi^{-1}(k)$ for all $i>k$. In this article, we count the number of connected permutation graphs on $n$ vertices with domination number $1$ and domination number $\frac{n}{2}$. We further show that for a natural number $k\leq \frac{n}{2}$, there exists a connected permutation graph on $n$ vertices with domination number $k$. We find a closed expression for the number of permutation graphs dominated by a set with two elements, and we find a closed expression for the number of permutation graphs efficiently dominated by any set of vertices. We conclude by providing an application of these results to strong fixed points, proving some conjectures posed on the OEIS.

中文翻译：

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置换图的支配数及其在强不动点上的应用

置换图 $G_\pi$ 是一个简单的图，其顶点对应于 $\pi$ 的元素，当 $i$ 和 $j$ 在 $\pi$ 中反转时，$i$ 和 $j$ 之间的边。当$G$ 中的每个顶点都是$D$ 的元素或与$D$ 的元素相邻时，一组顶点$D$ 被称为支配图$G$。支配数$\gamma(G)$被定义为$G$的最小支配集的基数。一个 $n$ 阶置换 $\pi$ 的强不动点是一个元素 $k$ 使得 $\pi^{-1}(j) \pi^{-1}(k)$ 对所有 $i > k$。在本文中，我们计算支配数为 $1$ 和支配数为 $\frac{n}{2}$ 的 $n$ 个顶点上的连通置换图的数量。我们进一步证明，对于自然数 $k\leq\frac{n}{2}$，在 $n$ 个顶点上存在一个连通的置换图，其支配数为 $k$。我们找到了由具有两个元素的集合支配的置换图数量的闭合表达式，并且我们找到了由任何顶点集有效支配的置换图数量的闭合表达式。最后，我们将这些结果应用于强不动点，证明了对 OEIS 提出的一些猜想。