Edgeless graphs are the only universal fixers

TL;DR

This paper proves that the only graphs that maintain their domination number under any permutation in the described construction are edgeless graphs, confirming a longstanding conjecture.

Contribution

The paper provides a complete proof that edgeless graphs are the only universal fixers, resolving a conjecture from 1999.

Findings

Edgeless graphs are the only universal fixers.

Confirmed the 1999 conjecture completely.

Established the invariance of domination number under permutations for edgeless graphs.

Abstract

Given two disjoint copies of a graph $G$, denoted $G^1$ and $G^2$, and a permutation $\pi$ of $V(G)$, the graph $\pi G$ is constructed by joining $u \in V(G^1)$ to $\pi(u) \in V(G^2)$ for all $u \in V(G^1)$. $G$ is said to be a universal fixer if the domination number of $\pi G$ is equal to the domination number of $G$ for all $\pi$ of $V(G)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. Since then, a few partial results have been shown. In this paper, we prove the conjecture completely.