We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $δ$ from each other. We investigate the complexity of this problem in terms of the rational parameter $δ$. The problem is polynomially solvable, if the numerator of $δ$ is $1$ or $2$, while all other cases turn out to be NP-hard.