Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles

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Abstract

According to the classical Erdős–Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most math formula vertices that hits all its cycles. Robertson and Seymour (J Comb Theory Ser B 41 (1986), 92–114) generalized this result in the best possible way. More specifically, they showed that if math formula is the class of all graphs that can be contracted to a fixed planar graph H, then every graph G either contains a set of k vertex-disjoint subgraphs of G, such that each of these subgraphs is isomorphic to some graph in math formula or there exists a set S of at most math formula vertices such that math formula contains no subgraph isomorphic to any graph in math formula. However, the function f is exponential. In this note, we prove that this function becomes quadratic when math formula consists all graphs that can be contracted to a fixed planar graph math formula. For a fixed c, math formula is the graph with two vertices and math formula parallel edges. Observe that for math formula this corresponds to the classical Erdős–Pósa theorem.

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