Planar Separators
Noga Alon, Paul D. Seymour, Robin B. Thomas · SIAM Journal on Discrete Mathematics · 1994
The authors give a short proof of a theorem of Lipton and Tarjan, that, for every planar graph with $n > 0$ vertices, there is a partition $( A,B,C )$ of its vertex set such that $|A|,|B| < \frac{2}{3}n,|C| \leq 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 deduce that, in fact, such a partition $( A,B,C )$ exists with $|A|,|B| < \frac{2}{3}n$, and $|C| \leq \frac{3}{2}( 2n )^{1/2} $ ; this improves the best previously known result. An analogous result holds when the vertices or edges are weighted.