A tight Erdős-Pósa function for planar minors

Abstract

Let H be a planar graph. By a classical result of Robertson and Seymour,

On the Erdős-Pósa property for long holes in $C_4$-free graphs

“…Let (A, v) be a pair considered in the i-th step of the second greedy packing such that Q = ∅, P is a path. We first show that V (P ) ∩ S 3 bd 20 (Q) = ∅. Suppose for a contradiction that P contains a vertex w of S 3 bd 20 (Q) .…”

“…Let a and b be almost C-dominating vertices and towards a contradiction suppose that a and b are not adjacent. Let P 1 , P 2 , P 3 be subpaths of C such that each P i has length 2 and dist C (P i , P j ) ≥ 2 for distinct i, j ∈ [3]. We can choose such paths because µ k ≥ 11 and |V (C)| > µ k .…”

“…Because X 5.1 ⊆ X 7.1 , by Lemma 5.2, H is not contained in S 3 Q . Let P be the connected component of H ∩ G[S 3 int 20 (Q) ] containing v. Then P is a path. Let Q be the other subpath of C with the same ends as Q.…”

“…In general, smaller bounding functions give improved approximation factors for the corresponding packing and covering problems. For example, the O(k log k) bounding function for planar minors proved in [3], currently gives the best approximation algorithm for packing planar minors.…”

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…Furthermore, for some problems including CYCLE PACKING, the Erdős-Pósa property gives an immediate parameterized algorithm. We refer the reader to a recent survey [139] and papers [119,140,141].…”