Planar separator theorem

Statement

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of $O(\sqrt{n})$ vertices from an $n$-vertex graph (where the $O$ invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most $2n/3$ vertices.