# 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

## Figures from this paper

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.

## References

SHOWING 1-10 OF 19 REFERENCES

• Computer Science
• 1971
Efficient algorithms are presented for partitioning a graph into connected components, biconnected components and simple paths. The algorithm for partitioning of a graph into simple paths is
• Mathematics, Computer Science
STOC '76
• 1976
It is shown that for each graph withn vertices and maximum in-degreed, there is a pebbling strategy which requires at mostc(d) n/logn pebbles, and this bound is tight to within a constant factor.
• Computer Science
• 1974
This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
This paper presents an unusual numbering of the mesh (unknowns) and shows that if the authors avoid operating on zeros, the $LDL^T$ factorization of A can be computed using the same standard algorithm in $O(n^3 )$ arithmetic operations.
• Computer Science
SIAM J. Comput.
• 1976
Classic binary search is extended to multidimensional search problems. This extension yields efficient algorithms for a number of tasks such as a secondary searching problem of Knuth, region location
• Computer Science
JACM
• 1976
Results are presented that establish hierarchies with respect to ≤S.T</italic></subscrpt> for (1) data structures, (2) sequential program schemata normal forms, and (3) sequential control structures.
• Computer Science
SWCT
• 1965
This paper investigates the computational complexity of binary sequences as measured by the rapidity of their generation by multitape Turing machines. A "translational" method which escapes some of
• Computer Science
JACM
• 1977
The context-sensitive languages cannot be recognized in linear time by deterministic multitape Turing machines, and are strictly contained in the class of languages recognized by Turing machines of tape complexity.