Strengthening Erdös-Pósa property for minor-closed graph classes

Let ℋ and 𝒢 be graph classes. We say that ℋ has the Erdős–Posa property for 𝒢 if for any graph G ∈𝒢, the minimum vertex covering of all ℋ-subgraphs of G is bounded by a function f of the maximum packing of ℋ-subgraphs in G (by ℋ-subgraph of G we mean any subgraph of G that belongs to ℋ). Robertson and Seymour [J Combin Theory Ser B 41 (1986), 92–114] proved that if ℋ is the class of all graphs that can be contracted to a fixed planar graph H, then ℋ has the Erdős–Posa property for the class of all graphs with an exponential bounding function. In this note, we prove that this function becomes linear when 𝒢 is any non-trivial minor-closed graph class. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:235-240, 2011 © 2011 Wiley Periodicals, Inc.

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