Packing and Covering Immersion Models of Planar Subcubic Graphs

  • Archontia C. Giannopoulou
  • O-joung Kwon
  • Jean-Florent Raymond
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9941)

Abstract

A graph H is an immersion of a graph G if H can be obtained by some subgraph G after lifting incident edges. We prove that there is a polynomial function \(f:{\mathbb {N}}\times {\mathbb {N}}\rightarrow {\mathbb {N}}\), such that if H is a connected planar subcubic graph on \(h>0\) edges, G is a graph, and k is a non-negative integer, then either G contains k vertex/edge-disjoint subgraphs, each containing H as an immersion, or G contains a set F of f(kh) vertices/edges such that \(G\setminus F\) does not contain H as an immersion.

Keywords

Erdö–Pósa properties Graph immersions Packings and coverings in graphs 

References

  1. 1.
    Belmonte, R., Giannopoulou, A., Lokshtanov, D., Thilikos, D.M.: The Structure of \(W_4\)-Immersion-Free Graphs. CoRR, abs/1602.02002 (2016)Google Scholar
  2. 2.
    Birmelé, E., Bondy, J.A., Reed, B.A.: The Erdős-Pósa property for long circuits. Combinatorica 27(2), 135–145 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chatzidimitriou, D., Raymond, J.-F., Sau, I., Thilikos, D.M.: Minors in graphs of large \(\theta _r\)-girth. CoRR, abs/1510.03041 (2015)Google Scholar
  4. 4.
    Chekuri, C., Chuzhoy, J.: Large-treewidth graph decompositions and applications. In: 45st Annual ACM Symposium on Theory of Computing (STOC), pp. 291–300 (2013)Google Scholar
  5. 5.
    Chekuri, C., Chuzhoy, J.: Polynomial bounds for the grid-minor theorem. CoRR, abs/1305.6577 (2013)Google Scholar
  6. 6.
    Chuzhoy, J.: Excluded grid theorem: improved and simplified. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, 14–17 June 2015, pp. 645–654 (2015)Google Scholar
  7. 7.
    Chuzhoy, J.: Improved bounds for the excluded grid theorem. CoRR, abs/1602.02629 (2015)Google Scholar
  8. 8.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  9. 9.
    Diestel, R., Kawarabayashi, K., Wollan, P.: The Erdős-Pósa property for clique minors in highly connected graphs. J. Comb. Theor. Ser. B 102(2), 454–469 (2012)CrossRefMATHGoogle Scholar
  10. 10.
    Ding, G., Oporowski, B.: On tree-partitions of graphs. Discrete Math. 149(1–3), 45–58 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Erdős, P., Pósa, L.: On independent circuits contained in a graph. Can. J. Math. 17, 347–352 (1965)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ganian, R., Kim, E.J., Szeider, S.: Algorithmic applications of tree-cut width. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9235, pp. 348–360. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  13. 13.
    Geelen, J., Kabell, K.: The Erdős-Pósa property for matroid circuits. J. Comb. Theor. Ser. B 99(2), 407–419 (2009)CrossRefMATHGoogle Scholar
  14. 14.
    Halin, R.: Tree-partitions of infinite graphs. Discrete Math. 97(1–3), 203–217 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kakimura, N., Kawarabayashi, K.: Fixed-parameter tractability for subset feedback set problems with parity constraints. Theor. Comput. Sci. 576, 61–76 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kawarabayashi, K.-I., Nakamoto, A.: The Erdös-pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces. Discrete Math. 307(6), 764–768 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Král’, D., Voss, H.-J.: Edge-disjoint odd cycles in planar graphs. J. Comb. Theor. Ser. B 90(1), 107–120 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, C.-H.: Packing and covering immersions in 4-edge-connected graphs. CoRR, abs/1505.00867 (2015)Google Scholar
  19. 19.
    Rautenbach, D., Reed, B.A.: The Erdos-Pósa property for odd cycles in highly connected graphs. Combinatorica 21(2), 267–278 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Raymond, J.-F., Sau, I., Thilikos, D.M.: An edge variant of the Erdős-Pósa property. Discrete Math. 339(8), 2027–2035 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Reed, B.A., Robertson, N., Seymour, P.D., Thomas, R.: Packing directed circuits. Combinatorica 16(4), 535–554 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Robertson, N., Seymour, P.D.: Graph minors. V. excluding a planar graph. J. Comb. Theor. Ser. B 41(2), 92–114 (1986)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Seese, D.: Tree-partite graphs and the complexity of algorithms. In: Budach, L. (ed.) Proceedings of Fundamentals of Computation Theory. LNCS, vol. 199, pp. 412–421. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  24. 24.
    Wollan, P.: The structure of graphs not admitting a fixed immersion. J. Comb. Theor. Ser. B 110, 47–66 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  • Archontia C. Giannopoulou
    • 1
  • O-joung Kwon
    • 2
  • Jean-Florent Raymond
    • 3
    • 5
  • Dimitrios M. Thilikos
    • 3
    • 4
  1. 1.Technische Universität BerlinBerlinGermany
  2. 2.Institute for Computer Science and ControlHungarian Academy of SciencesBudapestHungary
  3. 3.AlGCo Project-TeamCNRS, LIRMMMontpellierFrance
  4. 4.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  5. 5.University of WarsawWarsawPoland

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