# 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}
}
• Published 1 October 1977
• Mathematics
• Siam Journal on Applied Mathematics
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.
1,466 Citations
We show that every 2-connected triangulated planar graph with n vertices has a simple cycle C of length at most 4@@@@n which separates the interior vertices A from the exterior vertices B such that
• Mathematics
ArXiv
• 2018
We prove that a connected planar graph with $n$ vertices and $n+\mu$ edges has a vertex separator of size $O( \sqrt{\mu} + 1)$, and this separator can be computed in linear time.
• Mathematics
• 1990
Let G be an n-vertex graph with no minor isomorphic to an h- vertex complete 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
• Mathematics
• 2019
Definition 14.1 (Planar Graph). An undirected graph G is planar if it admits a planar drawing. A planar drawing is a drawing of G in the plane such that the vertices of G are points in the plane, and
An O(n5/4log n) algorithm is designed for finding the girth of an undirected n-vertex planar graph, giving the first o(n2) algorithm for this problem.
Two algorithms that have been developed recently by the author and his colleagues for designing a “sublinear-space” and polynomial-time separator algorithm are considered.
• Mathematics
MFCS
• 1988
It is shown that every planar graph with n vertices and a maximal degree k has an 0(√kn)-edge separator, and any n vertex tree can be divided into two parts of ≤ n / 2 vertices by removing 0(klog n/log k) edges.
• Mathematics
Acta Informatica
• 1997
If G is an n vertex maximal planar graph and δ≤1 3, then the vertex set of G can be partitioned into three sets A, B, C such that neither A nor B has weight exceeding 1−δ, and C is a simple cycle with no more than 2√n+O(1) vertices.