research-article

Computing excluded minors

Authors:

Stephan KreutzerAuthors Info & Claims

SODA '08: Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms

Pages 641 - 650

Published: 20 January 2008 Publication History

Abstract

By Robertson and Seymour's graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general method for obtaining such algorithms, which is based on definability in monadic second-order logic and the decidability of the monadic second-order theory of trees. A straightforward application of our method yields algorithms that, for a given k, compute excluded minor characterisations for the minor ideal T_k of all graphs of tree width at most k, the minor ideal B_k of all graphs of branch width at most k, and the minor ideal G_k of all graphs of genus at most k.

Our main results are concerned with constructions of new minor ideals from given ones. Answering a question that goes back to Fellows and Langston [11], we prove that there is an algorithm that, given excluded minor characterisations of two minor ideals C and D, computes such a characterisation for the ideal C ∪ D. Furthermore, we obtain an algorithm for computing an excluded minor characterisation for the class of all apex graphs over a minor ideal C, given an excluded minor characterisation for C. (An apex graph over C is a graph G from which one vertex can be removed to obtain a graph in C.) A corollary of this result is a uniform ftpalgorithm for the "distance k from planarity" problem.

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Cited By

Sau IStamoulis GThilikos D(2023) k-apices of minor-closed graph classes. I. Bounding the obstructionsJournal of Combinatorial Theory Series B10.1016/j.jctb.2023.02.012161:C(180-227)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1016/j.jctb.2023.02.012
Sau IStamoulis GThilikos D(2022)k-apices of Minor-closed Graph Classes. II. Parameterized AlgorithmsACM Transactions on Algorithms10.1145/351902818:3(1-30)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3519028
Leivaditis ASingh AStamoulis GThilikos DTsatsanis K(2021)Minor obstructions for apex-pseudoforestsDiscrete Mathematics10.1016/j.disc.2021.112529344:10Online publication date: 1-Oct-2021
https://dl.acm.org/doi/10.1016/j.disc.2021.112529
Show More Cited By

Index Terms

Computing excluded minors
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
    2. Graph theory
  2. Mathematical analysis
    1. Numerical analysis
      1. Computations in finite fields
2. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image ACM Conferences

SODA '08: Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms

January 2008

1289 pages

Program Chair:
Shang-Hua Teng
Boston University and Akamai Technologies, Inc.

Sponsors

SIAM Activity Group on Discrete Mathematics
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 20 January 2008

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SODA08

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SIGACT

SODA08: 19th ACM-SIAM Symposium on Discrete Algorithms

January 20 - 22, 2008

California, San Francisco

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

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Cited By

Sau IStamoulis GThilikos D(2023) k-apices of minor-closed graph classes. I. Bounding the obstructionsJournal of Combinatorial Theory Series B10.1016/j.jctb.2023.02.012161:C(180-227)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1016/j.jctb.2023.02.012
Sau IStamoulis GThilikos D(2022)k-apices of Minor-closed Graph Classes. II. Parameterized AlgorithmsACM Transactions on Algorithms10.1145/351902818:3(1-30)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3519028
Leivaditis ASingh AStamoulis GThilikos DTsatsanis K(2021)Minor obstructions for apex-pseudoforestsDiscrete Mathematics10.1016/j.disc.2021.112529344:10Online publication date: 1-Oct-2021
https://dl.acm.org/doi/10.1016/j.disc.2021.112529
Kawarabayashi KRossman BCzumaj A(2018)A polynomial excluded-minor approximation of treedepthProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175283(234-246)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175283
Giannopoulou AJansen BLokshtanov DSaurabh S(2017)Uniform Kernelization Complexity of Hitting Forbidden MinorsACM Transactions on Algorithms10.1145/302905113:3(1-35)Online publication date: 20-Mar-2017
https://dl.acm.org/doi/10.1145/3029051
Bulian JDawar A(2017)Fixed-Parameter Tractable Distances to Sparse Graph ClassesAlgorithmica10.1007/s00453-016-0235-779:1(139-158)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1007/s00453-016-0235-7
Bodlaender HFomin FLokshtanov DPenninkx ESaurabh SThilikos D(2016)(Meta) KernelizationJournal of the ACM10.1145/297374963:5(1-69)Online publication date: 8-Nov-2016
https://dl.acm.org/doi/10.1145/2973749
Fomin FLokshtanov DMisra NRamanujan MSaurabh SIndyk P(2015)Solving d-SAT via backdoors to small treewidthProceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722172(630-641)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722172
Cao YMarx D(2015)Interval Deletion Is Fixed-Parameter TractableACM Transactions on Algorithms10.1145/262959511:3(1-35)Online publication date: 13-Jan-2015
https://dl.acm.org/doi/10.1145/2629595
Cao YMarx DChekuri C(2014)Interval deletion is fixed-parameter tractableProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634083(122-141)Online publication date: 5-Jan-2014
https://dl.acm.org/doi/10.5555/2634074.2634083
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