The Planar Separator Theorem

The Planar Separator Theorem of Lipton and Tarjan [73] says that in any undirected planar graph G there exists a small separator \( S\; \subseteq {\rm{ }}V \) whose removal leaves two disjoint sets of vertices \( A,\;B \subseteq {\rm{ }}V \) with no edge between them; moreover, each of A and B is at most a constant fraction of the size of V.