Polynomial gap extensions of the Erdős-Pósa theorem

  • Jean-Florent Raymond
  • Dimitrios M. Thilikos
Conference paper
Part of the CRM Series book series (PSNS, volume 16)

Abstract

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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Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  • Jean-Florent Raymond
    • 1
  • Dimitrios M. Thilikos
    • 2
  1. 1.LIRMMMontpellierFrance
  2. 2.Department of MathematicsNational and Kapodistrian University of Athens and CNRS (LIRMM)France

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