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

TL;DR

This paper investigates the domination number of permutation graphs, counts graphs with specific domination numbers, and applies these findings to analyze strong fixed points, confirming conjectures from OEIS.

Contribution

It introduces new counts and formulas for domination numbers in permutation graphs and applies these to study strong fixed points, solving related conjectures.

Findings

Counted connected permutation graphs with domination number 1 and n/2.

Derived formulas for permutation graphs dominated by small vertex sets.

Connected domination numbers can range from 1 to n/2 in permutation graphs.

Abstract

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 $k\leq \frac{n}{2}$, 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.