Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T09:47:55.526Z Has data issue: false hasContentIssue false
  • English
  • Français

Excluded Forest Minors and the Erdős–Pósa Property

Published online by Cambridge University Press:  08 July 2013

SAMUEL FIORINI ,
GWENAËL JORET  and
DAVID R. WOOD
SAMUEL FIORINI
Affiliation:
Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium (e-mail: sfiorini@ulb.ac.be)
GWENAËL JORET
Affiliation:
Département d'Informatique, Université Libre de Bruxelles, Brussels, Belgium (e-mail: gjoret@ulb.ac.be)
DAVID R. WOOD
Affiliation:
School of Mathematical Sciences, Monash University, Melbourne, Australia (e-mail: david.wood@monash.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the so-called Erdős–Pósa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, the graph G has k vertex-disjoint subgraphs each containing H as a minor, or there exists a subset X of vertices of G with |X| ≤ f(k) such that G − X has no H-minor (see Robertson and Seymour, J. Combin. Theory Ser. B41 (1986) 92–114). While the best function f currently known is exponential in k, a O(k log k) bound is known in the special case where H is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour and Thomas on the pathwidth of graphs with an excluded forest-minor. In this paper we show that the function f can be taken to be linear when H is a forest. This is best possible in the sense that no linear bound is possible if H has a cycle.

Keywords

Primary 05C83Secondary 05C35
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 5 , September 2013 , pp. 700 - 721
Copyright
Copyright © The Author(s) 2013. Published by Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adler, I., Grohe, M. and Kreutzer, S. (2008) Computing excluded minors. In Proc. Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms, ACM, pp. 641–650.Google Scholar
[2]Bienstock, D., Robertson, N., Seymour, P. and Thomas, R. (1991) Quickly excluding a forest. J. Combin. Theory Ser. B 52 274–283.Google Scholar
[3]Bodlaender, H. L. and Kloks, T. (1996) Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21 358–402.Google Scholar
[4]Courcelle, B. (1990) Graph rewriting: An algebraic and logic approach. In Handbook of Theoretical Computer Science, Vol. B, Elsevier, pp. 193–242.Google Scholar
[5]Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M. and Saurabh, S. (2011) On the hardness of losing width. In Proc. 6th International Symposium on Parameterized and Exact Computation, Saarbrücken, Germany (Marx, D. and Rossmanith, P. eds), Vol. 7112 of Lecture Notes in Computer Science, Springer, pp. 159–168.Google Scholar
[6]Cygan, M., Pilipczuk, M., Pilipczuk, M. and Wojtaszczyk, J. (2010) An improved FPT algorithm and quadratic kernel for pathwidth one vertex deletion. In Parameterized and Exact Computation (Raman, V. and Saurabh, S., eds), Vol. 6478 of Lecture Notes in Computer Science, Springer, pp. 95–106.Google Scholar
[7]Diestel, R. (1995) Graph minors I: A short proof of the path-width theorem. Combin. Probab. Comput. 4 27–30.Google Scholar
[8]Diestel, R. (2010) Graph Theory, Vol. 173 of Graduate Texts in Mathematics, fourth edition, Springer.Google Scholar
[9]Diestel, R., Kawarabayashi, K. and Wollan, P. (2012) The Erdős–Pósa property for clique minors in highly connected graphs. J. Combin. Theory Ser. B 102 454–469.Google Scholar
[10]Ellis, J. A., Sudborough, I. H. and Turner, J. S. (1994) The vertex separation and search number of a graph. Inf. Comput. 113 50–79.CrossRefGoogle Scholar
[11]Erdős, P. and Pósa, L. (1965) On independent circuits contained in a graph. Canad. J. Math. 17 347–352.Google Scholar
[12]Fiorini, S., Joret, G. and Pietropaoli, U. (2010) Hitting diamonds and growing cacti. In Integer Programming and Combinatorial Optimization (Eisenbrand, F. and Shepherd, F., eds), Vol. 6080 of Lecture Notes in Computer Science, Springer, pp. 191–204.CrossRefGoogle Scholar
[13]Fomin, F. V., Lokshtanov, D., Misra, N. and Saurabh, S. (2012) Foundations of Computer Science (FOCS). In Proc. IEEE 53rd Annual Symposium on, Planar F-Deletion: Approximation, Kernelization and Optimal FPTAlgorithms, pp. 470–479.Google Scholar
[14]Fomin, F. V., Lokshtanov, D., Misra, N., Philip, G. and Saurabh, S. (2011) Hitting forbidden minors: Approximation and kernelization. In Proc. 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) (Schwentick, T. and Dürr, C., eds), Vol. 9, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 189–200.Google Scholar
[15]Fomin, F. V., Saurabh, S. and Thilikos, D. M. (2011) Strengthening Erdős–Pósa property for minor-closed graph classes. J. Graph Theory 66 235–240.CrossRefGoogle Scholar
[16]Grohe, M. (2008) Logic, graphs, and algorithms. In Logic and Automata: History and Perspectives (Flum, J., Grädel, E. and Wilke, T., eds), Amsterdam University Press, pp. 357–422.Google Scholar
[17]Grohe, M. and Marx, D. (2009) On tree width, bramble size, and expansion. J. Combin. Theory Ser. B 99 218–228.Google Scholar
[18]Grohe, M., Kawarabayashi, K., Marx, D. and Wollan, P. (2011) Finding topological subgraphs is fixed-parameter tractable. In Proc. 43rd Annual ACM Symposium on Theory of Computing: STOC '11, ACM, pp. 479–488.CrossRefGoogle Scholar
[19]Joret, G., Paul, C., Sau, I., Saurabh, S. and Thomassé, S. (2011) Hitting and harvesting pumpkins. In Algorithms: ESA 2011 (Demetrescu, C. and Halldórsson, M., eds), Vol. 6942 of Lecture Notes in Computer Science, Springer, pp. 394–407.CrossRefGoogle Scholar
[20]Kawarabayashi, K. and Kobayashi, Y. (2012) Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid. In Proc. 29th International Symposium on Theoretical Aspects of Computer Science: STACS 2012 (Dürr, C. and Wilke, T., eds), Vol. 14 of Leibniz International Proceedings in Informatics, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, pp. 278–289.Google Scholar
[21]Kim, E. J., Paul, C. and Philip, G. A single-exponential FPT algorithm for the K 4-minor cover problem. In Proc. 13th Scandinavian Symposium and Workshops on Algorithm Theory: SWAT 2012, to appear. arXiv:1204.1417Google Scholar
[22]Lagergren, J. (1998) Upper bounds on the size of obstructions and intertwines. J. Combin. Theory Ser. B 73 7–40.Google Scholar
[23]Leaf, A. and Seymour, P. Treewidth and planar minors. Preprint. http://web.math.princeton.edu/~pds/papers/treewidth/paper.pdfGoogle Scholar
[24]Lubotzky, A., Phillips, R. and Sarnak, P. (1988) Ramanujan graphs. Combinatorica 8 261–277.Google Scholar
[25]Philip, G., Raman, V. and Villanger, Y. (2010) A quartic kernel for pathwidth-one vertex deletion. In Graph Theoretic Concepts in Computer Science (Thilikos, D., ed.), Vol. 6410 of Lecture Notes in Computer Science, Springer, pp. 196–207.Google Scholar
[26]Robertson, N. and Seymour, P. D. (1986) Graph minors V: Excluding a planar graph. J. Combin. Theory Ser. B 41 92–114.Google Scholar
[27]Robertson, N. and Seymour, P. D. (2004) Graph minors XX: Wagner's conjecture. J. Combin. Theory Ser. B 92 325–357.Google Scholar
[28]Robertson, N., Seymour, P. and Thomas, R. (1994) Quickly excluding a planar graph. J. Combin. Theory Ser. B 62 323–348.Google Scholar
[29]Seese, D. (1991) The structure of the models of decidable monadic theories of graphs. Ann. Pure Appl. Logic 53 169–195.Google Scholar
[30]Thatcher, J. W. and Wright, J. B. (1968) Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Systems Theory 2 57–81.Google Scholar