research-article

An excluded grid theorem for digraphs with forbidden minors

Authors:

Ken-ichi Kawarabayashi,

Stephan KreutzerAuthors Info & Claims

SODA '14: Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms

Pages 72 - 81

Published: 05 January 2014 Publication History

Abstract

The excluded grid theorem, originally proved by Robertson and Seymour in Graph Minors V, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance as the basis for bidimensionality theory on graph classes excluding a fixed minor.

In 1997, Reed [22] and later Johnson, Robertson, Seymour and Thomas [16] conjectured an analogous theorem for directed graphs, i.e. the existence of a function f: N → N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas gave a proof of this conjecture for planar digraphs but no result beyond planar graphs is known to date.

In this paper we prove the conjecture for the case of digraphs excluding a fixed undirected graph as a minor. For algorithmic applications our theorem is particularly interesting as it covers those classes of digraphs to which, on undirected graphs, theories based on the excluded grid theorem such as bidimensionality theory apply. We expect similar applications for directed graphs in particular to algorithmic versions of Erdős-Pósa type results and the directed disjoint paths problem.

References

[1]

I. Adler, S. G. Kolliopoulos, P. K. Krause, D. Lokshtanov, S. Saurabh, and D. M. Thilikos. Tight bounds for linkages in planar graphs. In ICALP, pages 110--121, 2011.

Digital Library

[2]

H. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth. Technical report, Utrecht University, 1996. Original article appeared in SIAM Journal on Computing, Volume 25, 1996.

Digital Library

[3]

H. L. Bodlaender. Treewidth: Algorithmic techniques and results. In Proc. of Mathematical Foundations of Computer Science (MFCS), volume 1295 of Lecture Notes in Computer Science, pages 19--36, 1997.

Digital Library

[4]

H. L. Bodlaender. Discovering treewidth. In 31st International Conference on Current Trends in Theory and Practice of Computer Science, pages 1--16, 2005.

Digital Library

[5]

C. Chekuri and J. Chuzhoy. Polynomial Bounds for the Grid-Minor Theorem. unpublished manuscript, 2013.

[6]

M. Cygan, D. Marx, M. Pilipczuk, and M. Pilipczuk. The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. arXiv (CoRR), abs/1304.4207, 2013.

[7]

E. Demaine and M. Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, pages 332--337, 2008.

Digital Library

[8]

E. Demaine and M. Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1): 19--36, 2008.

Digital Library

[9]

E. D. Demaine and M. T. Hajiaghayi. Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth. In Graph Drawing, pages 517--533, 2004.

Digital Library

[10]

E. D. Demaine and M. T. Hajiaghayi. Bidimensionality: new connections between FPT algorithms and PTASs. In SODA, pages 590--601, 2005.

Digital Library

[11]

R. Diestel. Graph Theory. Springer-Verlag, 3rd edition, 2005.

[12]

R. Downey and M. Fellows. Parameterized Complexity. Springer, 1998.

[13]

P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2: 463--470, 1935.

[14]

F. V. Fomin, D. Lokshtanov, V. Raman, and S. Saurabh. Bidimensionality and EPTAS. In SODA, pages 748--759, 2011.

Digital Library

[15]

F. V. Fomin, D. Lokshtanov, S. Saurabh, and D. M. Thilikos. Bidimensionality and kernels. In SODA, pages 503--510, 2010.

Digital Library

[16]

T. Johnson, N. Robertson, P. D. Seymour, and R. Thomas. Directed tree-width. J. Comb. Theory, Ser. B, 82(1): 138--154, 2001.

Digital Library

[17]

T. Johnson, N. Robertson, P. D. Seymour, and R. Thomas. Excluding a grid minor in digraphs. unpublished manuscript, 2001.

[18]

K. Kawarabayashi and Y. Kobayashi. Linear min-max relation between the treewidth of h-minor-free graphs and its largest grid. In C. Dürr and T. Wilke, editors, STACS, volume 14 of LIPIcs, pages 278--289. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2012.

[19]

K. Kawarabayashi, Y. Kobayashi, and B. Reed. The disjoint paths problem in quadratic time. J. Combin. Theory Ser. B, 102: 424--435, 2012.

Digital Library

[20]

K. Kawarabayashi, M. Krčál, D. Král, and S. Kreutzer. Packing directed cycles through a specified vertex set. In ACM-SIAM Symp. on Discrete Algorithms (SODA), 2013.

Digital Library

[21]

S. Kreutzer and S. Tazari. Directed nowhere dense classes of graphs. In Proc. of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), 2012.

Digital Library

[22]

B. Reed. Tree width and tangles: A new connectivity measure and some applications. In R. Bailey, editor, Surveys in Combinatorics, pages 87--162. Cambridge University Press, 1997.

[23]

B. Reed. Introducing directed tree-width. Electronic Notes in Discrete Mathematics, 3: 222--229, 1999.

[24]

B. A. Reed, N. Robertson, P. D. Seymour, and R. Thomas. Packing directed circuits. Combinatorica, 16(4): 535--554, 1996.

[25]

B. A. Reed and D. R. Wood. Polynomial treewidth forces a large grid-like-minor. Eur. J. Comb., 33(3): 374--379, 2012.

Digital Library

[26]

N. Robertson and P. Seymour. Graph minors I -- XXIII, 1982--2010. Appearing in Journal of Combinatorial Theory, Series B from 1982 till 2010.

[27]

N. Robertson and P. Seymour. Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B, 63: 65--110, 1995.

Digital Library

[28]

N. Robertson and P. Seymour. Graph minors XVI. Excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 77: 1--27, 1999.

Digital Library

[29]

N. Robertson, P. Seymour, and R. Thomas. Quickly excluding a planar graph. Journal of Combinatorial Theory, Series B, 62: 323--348, 1994.

Digital Library

[30]

N. Robertson and P. D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1): 92--114, 1986.

Digital Library

[31]

P. Seymour. Disjoint paths in graphs. Discrete Math., 29: 293--309, 1980.

[32]

C. Thomassen. 2-linked graphs. European Journal of Combinatorics, 1: 371--378, 1980.

Cited By

Kawarabayashi KKreutzer SServedio RRubinfeld R(2015)The Directed Grid TheoremProceedings of the forty-seventh annual ACM symposium on Theory of Computing10.1145/2746539.2746586(655-664)Online publication date: 14-Jun-2015
https://dl.acm.org/doi/10.1145/2746539.2746586
Kawarabayashi KKreutzer S(2015)Towards the Graph Minor Theorems for Directed GraphsProceedings, Part II, of the 42nd International Colloquium on Automata, Languages, and Programming - Volume 913510.1007/978-3-662-47666-6_1(3-10)Online publication date: 6-Jul-2015
https://dl.acm.org/doi/10.1007/978-3-662-47666-6_1
Bojanczyk MDittmann CKreutzer SHenzinger TMiller D(2014)Decomposition theorems and model-checking for the modal μ-calculusProceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1145/2603088.2603144(1-10)Online publication date: 14-Jul-2014
https://dl.acm.org/doi/10.1145/2603088.2603144
Show More Cited By

Index Terms

An excluded grid theorem for digraphs with forbidden minors
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Graph enumeration
2. Theory of computation
  1. Design and analysis of algorithms

Recommendations

An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem
STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

The excluded grid theorem, originally proved by Robertson and Seymour in Graph Minors V, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance as the basis ...
Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs
SOFSEM 2019: Theory and Practice of Computer Science
Abstract
There have been many attempts to find directed graph classes with bounded directed path-width and bounded directed tree-width. Right now, the only known directed tree-width-/path-width-bounded graphs are cycle-free graphs with directed path-width ...
Coloring hypergraphs with excluded minors
Abstract
Hadwiger’s conjecture, among the most famous open problems in graph theory, states that every graph that does not contain K t as a minor is properly ( t − 1 )-colorable.

The purpose of this work is to demonstrate that a natural extension of ...

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences

SODA '14: Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms

January 2014

1910 pages

ISBN:9781611973389

Program Chair:
Chandra Chekuri
University of Illinois, Urbana-Champaign

Sponsors

SIAM Activity Group on Discrete Mathematics

In-Cooperation

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 05 January 2014

Check for updates

Qualifiers

Research-article

Conference

SODA '14

Sponsor:

SODA '14: ACM-SIAM Symposium on Discrete Algorithms

January 5 - 7, 2014

Oregon, Portland

Acceptance Rates

Overall Acceptance Rate 411 of 1,322 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
49
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Kawarabayashi KKreutzer SServedio RRubinfeld R(2015)The Directed Grid TheoremProceedings of the forty-seventh annual ACM symposium on Theory of Computing10.1145/2746539.2746586(655-664)Online publication date: 14-Jun-2015
https://dl.acm.org/doi/10.1145/2746539.2746586
Kawarabayashi KKreutzer S(2015)Towards the Graph Minor Theorems for Directed GraphsProceedings, Part II, of the 42nd International Colloquium on Automata, Languages, and Programming - Volume 913510.1007/978-3-662-47666-6_1(3-10)Online publication date: 6-Jul-2015
https://dl.acm.org/doi/10.1007/978-3-662-47666-6_1
Bojanczyk MDittmann CKreutzer SHenzinger TMiller D(2014)Decomposition theorems and model-checking for the modal μ-calculusProceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1145/2603088.2603144(1-10)Online publication date: 14-Jul-2014
https://dl.acm.org/doi/10.1145/2603088.2603144
Kawarabayashi KKobayashi YKreutzer SShmoys D(2014)An excluded half-integral grid theorem for digraphs and the directed disjoint paths problemProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591876(70-78)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591876

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents