# A Separator Theorem for Planar Graphs

@article{Lipton1977AST, title={A Separator Theorem for Planar Graphs}, author={Richard J. Lipton and Robert Endre Tarjan}, journal={Siam Journal on Applied Mathematics}, year={1977}, volume={36}, pages={177-189} }

Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.

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