  We introduce a new balanced separator theorem for unit-disk graphs involving
two shortest paths combined with the 1-hop neighbours of those paths and two
other vertices. This answers an open problem of Yan, Xiang and Dragan [CGTA
'12] and improves their result that requires removing the 3-hop neighborhood of
two shortest paths. Our proof uses very different ideas, including Delaunay
triangulations and a generalization of the celebrated balanced separator
theorem of Lipton and Tarjan [J. Appl. Math. '79] to systems of
non-intersecting paths.
