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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 2 n vertices. We exhibit an algorithm which finds such a partition A, B, C in O (n) time.
Lipton et al. (Sun,) studied this question.