A Tight Erdös--Pósa Function for Wheel Minors

Abstract

Let W t denote the wheel on t + 1 vertices. We prove that for every integer t ≥ 3 there is a constant c = c(t) such that for every integer k ≥ 1 and every graph G, either G has k vertex-disjoint subgraphs each containing W t as a minor, or there is a subset X of at most ck log k vertices such that G − X has no W t minor. This is best possible, up to the value of c. We conjecture that the result remains true more generally if we replace W t with any fixed planar graph H.

A tight Erdős-Pósa function for planar minors

“…Then there is a function s 2. 5 and an algorithm that, given an n-vertex graph G and an integer k, decides whether π(G) ≤ k in time 2 s 2.5 (t)k log k + h(s 2.5 (t)k log k) n O (1) .…”

“…A part is the spine or a leg of C R . If some part of C R contains at least √ y matched vertices, then (1) or (3) holds, and we are done.…”

“…Then G−(X ∪Y ) has no cycle of length 0 mod m, and |X ∪Y | ≤ ck log(k +1)+2m(k −1) = O(k log k), as desired. We will also need the following more precise version of Lemma 5.3, which appears in [1].…”

“…build on their approach. Our main technical contribution is a series of lemmas which allowed us to develop the 'right' generalization of the objects used in [1]. An overview of the proof will be given shortly but first let us mention some combinatorial and algorithmic consequences of our result.…”

A tight Erdős-Pósa function for planar minors

“…Prior to this paper, when H is planar but not a forest, a O(k log k) bounding function was known to hold for H-models if H is a triangle [15], a cycle [17,39], a multigraph consisting of two vertices linked by parallel edges [9], and more generally if H is any minor of a wheel [1]. The authors of [1] developed general tools to tackle arbitrary planar graphs H, together with some techniques that are specific to wheels. In this paper we build on their approach.…”