On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points

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 $j<k$, and ${\pi }^{-1}(i)>{\pi }^{-1}(k)$ 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 $k\le \frac{n}{2}$, 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.