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.

翻译：本文针对单位圆盘图提出了一种新的平衡分隔子定理，该定理结合了两条最短路径及其1跳邻域顶点与另外两个顶点。这解决了Yan、Xiang与Dragan在[CGT A '12]中提出的公开问题，并改进了他们需要移除两条最短路径3跳邻域的结果。我们的证明采用了截然不同的思路，包括Delaunay三角剖分技术，并将Lipton与Tarjan在[J. Appl. Math. '79]中提出的经典平衡分隔子定理推广至非相交路径系统。