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

According to the classical Erd˝´ theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most O(klogk) vertices that hits all its cycles. Robertson and Seymour [Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B, 41:92‐114, 1986] generalized this result in the best possible way. More specifically, they showed that ifH 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 inH or there exists a set S of at most f(k) vertices such that Gn S contains no subgraph isomorphic to any graph inH. However the function f is exponential. In this note, we prove that this function becomes quadratic whenH consists all graphs that can be contracted to a fixed planar graph c. For a fixed c, c is the graph with two vertices and c 1 parallel edges. Observe that for c = 2 this corresponds to the classical Erd˝´ osa theorem.

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