  We continue the study of $\delta$-dispersion, 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 every two facilities
have distance at least $\delta$ from each other. Our main technical
contribution is an efficient procedure to `round-up' distance $\delta$. It
transforms a $\delta$-dispersed set $S$ into a $\delta^\star$-dispersed set
$S^\star$ of same size where distance $\delta^\star$ is a slightly larger
rational $\tfrac{a}{b}$ with a numerator $a$ upper bounded by the longest
(not-induced) path in the input graph. Based on this rounding procedure and
connections to the distance-$d$ independent set problem we derive a number of
algorithmic results. When parameterized by treewidth, the problem is in XP.
When parameterized by treedepth the problem is FPT and has a matching lower
bound on its time complexity under ETH. Moreover, we can also settle the
parameterized complexity with the solution size as parameter using our rounding
technique: $\delta$-\dispersion is FPT for every $\delta \leq 2$ and W[1]-hard
for every $\delta > 2$. Further, we show that $\delta$-dispersion is
NP-complete for every fixed irrational distance $\delta$, which was left open
in a previous work.
