Polynomial gap extensions of the Erdős-Pósa theorem
Given a graph H, we denote by M(H) all graphs that can be contracted to H. The following extension of the Erdős-Pósa Theorem holds:for every h-vertex planar graph H, there exists a function fH: N → N such that every graph G, either contains k disjoint copies of graphs in M(H), or contains a set of fH(k) vertices meeting every subgraph of G that belongs in M(H). In this paper we prove that fH can be polynomially (upper) bounded for every graph H of pathwidth at most 2 and, in particular, that fH(k) = 2o(h2). k2. log k. As a main ingredient of the proof of our result, we show that for every graph H on h vertices and pathwidth at most 2, either G contains k disjoint copies of H as a minor or the treewidth of G is upper-bounded by 2o(h)2. k2. log k. We finally prove that the exponential dependence on h in these bounds can be avoided if H = K2,r. In particular, we show that fK2,r = O (r2. k2).
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- C. Chekuri and J. Chuzhoy, Large-treewidth graph decompositions and applications, In:“45st Annual ACM Symposium on Theory of Computing”, (STOC 2013), 2013.Google Scholar
- R. Diestel, “Graph Theory”, volume 173 of Graduate Texts in Mathematics, Springer-Verlag, Heidelberg, fourth edition, 2010.Google Scholar
- P. Erdős and G. Szekeres, A combinatorial problem in geometry, In:“Classic Papers in Combinatorics”, Ira Gessel and Gian-Carlo Rota (eds.), Modern Birkhäuser Classics, Birkhäuser Boston, 1987, 49–56.Google Scholar
- S. Fiorini, T. Huyhn and G. Joret, personal communication, 2013.Google Scholar
- S. Fiorini, G. Joret and I. Sau, Optimal Erdős-Pósa property for pumpkins, Manuscript, 2013.Google Scholar
- F. V. Fomin, D. Lokshtanov, N. Misra, G. Philip and S. Saurabh, Quadratic upper bounds on the Erdős-Pósa property for a generalization of packing and covering cycles, Journal of Graph Theory, to appear in 2013.Google Scholar
- Ken-ichi Kawarabayashi and Y. Kobayashi, Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid, In:“29th Int. Symposium on Theoretical Aspects of Computer Science (STACS 2012)”, Vol. 14 of LIPIcs, Dagstuhl, Germany, 2012, 278–289.Google Scholar
- A. Leaf and P. Seymour, Treewidth and planar minors, Manuscript, 2012.Google Scholar
- D. R. Wood Samuel Fiorini and G. Joret, Excluded forest minors and the Erdős-Pósa property, Technical report, Cornell University, 2012.Google Scholar