Planar Separators

The authors give a short proof of a theorem of Lipton and Tarjan, that, for every planar graph with n > 0 vertives, there is a partition (A, B, C,) of its vertex set such that $|A|, |B| < \frac{2}{3}n, |C| \geq 2(2n)^{1/2},$ and no vertex in A is adjacent to any vertex in B. Secondly, they apply the same technique more carefully to deduced that, in fact, such a partition (A, B, C) exists with $|A|, |B| < \frac{2}{3}n, |C| \geq 2(2n)^{1/2};$ this improves the best previously known result. An analogous result holds when the vertices or edges are weighted.