# A linear-time algorithm to find a separator in a graph excluding a minor

@article{Reed2009ALA, title={A linear-time algorithm to find a separator in a graph excluding a minor}, author={B. Reed and D. Wood}, journal={ACM Trans. Algorithms}, year={2009}, volume={5}, pages={39:1-39:16} }

Let <i>G</i> be an <i>n</i>-vertex <i>m</i>-edge graph with weighted vertices. A pair of vertex sets <i>A</i>, <i>B</i> ⊆ <i>V</i>(<i>G</i>) is a 2/3<i>-separation</i> of <i>order</i> |<i>A</i> ∩ <i>B</i>| if <i>A</i> ∪ <i>B</i> = <i>V</i>(<i>G</i>), there is no edge between <i>A</i> − <i>B</i> and <i>B</i> − <i>A</i>, and both <i>A</i> − <i>B</i> and <i>B</i> − <i>A</i> have weight at most 2/3 the total weight of <i>G</i>. Let ℓ ∈ Z<sup>+</sup> be fixed. Alon et al. [1990] presented an… Expand

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#### References

SHOWING 1-10 OF 79 REFERENCES

Maximum matching in graphs with an excluded minor

- Mathematics, Computer Science
- SODA '07
- 2007

A randomized algorithm that extends a previous randomized algorithm of Mucha and Sankowski, having the same running time, that finds a maximum matching in a planar graphs and a deterministic algorithm with a running time of 2.19 seconds for counting the number of perfect matchings in graphs with bounded genus. Expand

Constant factor approximation of vertex-cuts in planar graphs

- Mathematics, Computer Science
- STOC '03
- 2003

A structural theorem is proved for planar graphs, showing the existence of a near-optimal quotient vertex-cut whose high-level structure is that of a bounded-depth tree and an algorithm is developed that optimizes over such complex structures in running time that depends (exponentially) not on the size of the structure, but rather only on its depth. Expand

Separators for sphere-packings and nearest neighbor graphs

- Mathematics, Computer Science
- JACM
- 1997

This result implies that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, which gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. Expand

Reduced constants for simple cycle graph separation

- Computer Science, 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. Expand

A separator theorem for graphs with an excluded minor and its applications

- Mathematics, Computer Science
- STOC '90
- 1990

It follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/ √ log n in polynomial time, find that size exactly and solve any sparse system of n linear equations in n unknowns in time O(n). Expand

Divide-and-conquer approximation algorithms via spreading metrics

- Mathematics, Computer Science
- JACM
- 2000

A polynomial time approximation algorithm is presented for problems modeled by the divide-and-conquer paradigm, such that all subgraphs for which the optimization problem is nontrivial have large diameters. Expand

Finding small balanced separators

- Mathematics, Computer Science
- STOC '06
- 2006

This paper gives a randomized algorithm that finds an α-separator of size k in the given graph, unless the graph contains an (α+ε)-separators of size strictly less than k, in which case the algorithm finds one such separator. Expand

A Separator Theorem for Graphs of Bounded Genus

- Mathematics, Computer Science
- J. Algorithms
- 1984

The main result of this paper is that if the authors can draw a graph on a surface of genus g, then they can bisect it by removing $O(\sqrt{gn})$ vertices, best possible to within a constant factor. Expand

Separators in Graphs with Negative and Multiple Vertex Weights

- Computer Science, Mathematics
- Algorithmica
- 1999

It is shown that if the vertices of the graph have real-valued weights, which may be positive or negative, then the graph can be divided exactly in half according to weight. Expand

Planar Separators

- Physics, Computer Science
- SIAM J. Discret. Math.
- 1994

The authors give a short proof of a theorem of Lipton and Tarjan, that, for every planar graph with n > 0 vertives, there is a partition (A, B, C,) of its vertex set such that no vertex in A is adjacent to any vertex in B. Expand