  We prove a geometric version of the graph separator theorem for the unit disk
intersection graph: for any set of $n$ unit disks in the plane there exists a
line $\ell$ such that $\ell$ intersects at most $O(\sqrt{(m+n)\log{n}})$ disks
and each of the halfplanes determined by $\ell$ contains at most $2n/3$ unit
disks from the set, where $m$ is the number of intersecting pairs of disks. We
also show that an axis-parallel line intersecting $O(\sqrt{m+n})$ disks exists,
but each halfplane may contain up to $4n/5$ disks. We give an almost tight
lower bound (up to sublogarithmic factors) for our approach, and also show that
no line-separator of sublinear size in $n$ exists when we look at disks of
arbitrary radii, even when $m=0$. Proofs are constructive and suggest simple
algorithms that run in linear time. Experimental evaluation has also been
conducted, which shows that for random instances our method outperforms the
method by Fox and Pach (whose separator has size $O(\sqrt{m})$).
