  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.
