## A separator theorem for graphs with an excluded minor and its applications (1990)

### Cached

### Download Links

- [www.cs.tau.ac.il]
- [www.math.tau.ac.il]
- [www.math.tau.ac.il]
- [www.tau.ac.il]
- [www.cs.tau.ac.il]
- DBLP

### Other Repositories/Bibliography

Venue: | IN PROCEEDINGS OF THE 22ND ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING |

Citations: | 93 - 1 self |

### BibTeX

@INPROCEEDINGS{Alon90aseparator,

author = {Noga Alon and Paul Seymour and Robin Thomas},

title = {A separator theorem for graphs with an excluded minor and its applications},

booktitle = {IN PROCEEDINGS OF THE 22ND ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING},

year = {1990},

pages = {293--299},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Let G be an n-vertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given h-vertex graph H. We prove that there is a set X of no more than h 3/2 n 1/2 vertices of G whose deletion creates a graph in which the total weight of every connected component is at most 1/2. This extends significantly a well-known theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds, given an n-vertex graph G with weights as above and an h-vertex graph H, either such a set X or a minor of G isomorphic to H. The algorithm runs in time O(h 1/2 n 1/2 m), where m is the number of edges of G plus the number of its vertices. Our results supply extensions of the many known applications of the Lipton-Tarjan separator theorem from the class of planar graphs (or that of graphs with bounded genus) to any class of graphs with an excluded minor. For example, 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 find the chromatic number of G in time 2 O( √ n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure 0 corresponds to G in time O(n 3/2). We also describe a combinatorial application of our result which relates the tree-width of a graph to the maximum size of a Kh-minor in it.

### Citations

1371 |
Graph Theory with Applications
- Bondy, Murty
- 1976
(Show Context)
Citation Context ...raph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. By an H-minor of G we mean a minor of G isomorphic to H. Thus, the Kuratowski-Wagner Theorem (see, e.g., =-=[1]-=- ) asserts that planar graphs are those without K5 or K3,3 minors. We prove the following result. Theorem 1.2 Let h ≥ 1 be an integer, let H be a simple graph with h vertices, and let G be a graph wit... |

401 | A separator theorem for planar graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ...a graph G is a pair (A, B) of subsets of V (G) with A ∪ B = V (G), such that no edge of G joins a vertex in A − B to a vertex in B − A. Its order is |A ∩ B|. A well-known theorem of Lipton and Tarjan =-=[7]-=- asserts the following. ( R + denotes the set of non-negative real numbers. If w : V (G) → R + is a function and X ⊆ V (G), we denote � (w(v) : v ∈ X) by w(X).) Theorem 1.1 (Lipton-Tarjan [7]) Let G b... |

190 |
Generalized nested dissection
- Lipton, Rose, et al.
- 1979
(Show Context)
Citation Context ...ct result. Theorem 1.5 Let h ≥ 1 be an integer, and let G be a graph with n vertices with a haven of order h 3/2 n 1/2 . Then G has a Kh-minor. Lipton and Tarjan [8] , [9] and Lipton, Rose and Tarjan =-=[6]-=- gave many applications of the planar separator theorem (and noted that most of them would generalize to any family of graphs with small separators.) Indeed our results supply simple generalizations o... |

171 |
Applications of a planar separator theorem
- Lipton, Tarjan
- 1980
(Show Context)
Citation Context ... following more general and more compact result. Theorem 1.5 Let h ≥ 1 be an integer, and let G be a graph with n vertices with a haven of order h 3/2 n 1/2 . Then G has a Kh-minor. Lipton and Tarjan =-=[8]-=- , [9] and Lipton, Rose and Tarjan [6] gave many applications of the planar separator theorem (and noted that most of them would generalize to any family of graphs with small separators.) Indeed our r... |

84 |
A separator theorem for graphs of bounded genus
- Gilbert, Hutchinson, et al.
- 1984
(Show Context)
Citation Context ...th this case. We suspect that the estimate h3/2n1/2 in Theorem 1.2 can be replaced by O(hn1/2 ). Since the genus of Kh is Θ(h2 ) this would extend, if true, a result of Gilbert, Hutchinson and Tarjan =-=[3]-=- who proved that any graph on n vertices with genus g has a separator of order O(g1/2n1/2 ). Although an O(hn1/2 ) result would be an asymptotically better result than the one stated above, an advanta... |

79 |
An extremal function for contractions of graphs
- Thomason
- 1984
(Show Context)
Citation Context ...pendent set in each piece by exhaustive search, and combine the results to obtain the desired approximation. To estimate the relative error here one can apply the result of Kostochka [4] and Thomason =-=[12]-=- and a simple greedy-argument to obtain a lower bound for the size of the maximum independent set in an n-vertex graph with no Kh-minor. This gives: Proposition 4.2 There is an algorithm that approxim... |

59 |
The graph genus problem is NP-complete
- Thomassen
- 1989
(Show Context)
Citation Context ...n time which is polynomial in both the genus and the size of the graph, and in fact this is impossible if P �= NP , since the problem of determining the genus of a graph is, as proved by C. Thomassen =-=[11]-=-, NP-complete. Moreover, even for bounded genus, the best known algorithm for finding such an embedding is much slower than our separator algorithm). Although our algorithm is not as efficient, it is ... |

51 |
An observation on time-storage trade off
- Cook
- 1973
(Show Context)
Citation Context ...ximum independent set in an n-vertex graph with a fixed excluded minor with relative error of O( 1 log n ). Pebbling is a one person game that arises in the study of time-space trade-offs (see, e.g., =-=[2]-=-). An immediate extension of a result from [9] gives; Proposition 4.4 Any n-vertex acyclic digraph with no Kh-minor and with maximum in-degree k can be pebbled using O(h 3/2 n 1/2 + k log n) pebbles. ... |

11 |
Graph Searching and a Minimax Theorem for Treewidth
- Seymour, Thomas
(Show Context)
Citation Context ...Wt1 ∩ Wt3 ⊆ Wt2. The tree-width of G is the minimum k such that there is a tree-decomposition (T, W ) of T satisfying |Wt| ≤ k + 1 for all t ∈ V (T ). Combining Theorem 1.5 with one of the results of =-=[10]-=- we can prove the following. Proposition 4.5 Let h ≥ 1 be an integer, and let G be a graph with n vertices and with tree-width at least h 3/2 n 1/2 . Then G has a Kh-minor. Acknowledgement We would li... |

10 |
A lower bound for the hadwiger number of a graph as a function of the average degree of its vertices, Discret
- Kostochka
- 1982
(Show Context)
Citation Context ... the maximum independent set in each piece by exhaustive search, and combine the results to obtain the desired approximation. To estimate the relative error here one can apply the result of Kostochka =-=[4]-=- and Thomason [12] and a simple greedy-argument to obtain a lower bound for the size of the maximum independent set in an n-vertex graph with no Kh-minor. This gives: Proposition 4.2 There is an algor... |

2 |
Area-efficient graph layouts (for
- Leiserson
- 1980
(Show Context)
Citation Context ...an be found. At the moment this remains an open problem. 4 Applications Efficient algorithms for finding small separators in graphs are useful in the layout of circuits in a model of VLSI (see, e.g., =-=[5]-=-). Thus our results can be applied for finding efficiently embeddings of graphs with excluded minors. All the applications of the Lipton-Tarjan planar separator theorem given in [9] and in [6] carry o... |

1 |
Nonserial dynamic programming is optimal
- Rosenthal
- 1977
(Show Context)
Citation Context .../2 (log n) 1/2 ) in time O(h1/2n1/2m). Separator theorems are useful in designing efficient divide-and-conquer algorithms. An example given in [9] is that of nonserial dynamic programming (see, e.g., =-=[13]-=-). Proposition 4.3 Let f(x1, . . . , xn) be a function of n variables, where each variable takes values in a finite set S of s elements, and suppose f is a sum of functions fi, where each fi is a func... |