Improved Bounds for the Flat Wall Theorem

TL;DR

This paper improves bounds related to the Flat Wall Theorem, providing tighter parameters for the existence of large flat walls or $K_t$-minors in graphs, with applications to algorithms and matching lower bounds.

Contribution

It presents significantly improved bounds for the Flat Wall Theorem and extends results to graphs with bounded degree, including efficient algorithms and lower bounds.

Findings

Improved bounds: $f(w,t)= heta(t(t+w))$ with small vertex set $A$

Special case: graphs with max degree $D$ contain large flat walls or $K_t$-minors

Provided algorithms to find either a $K_t$-minor or a flat wall

Abstract

The Flat Wall Theorem of Robertson and Seymour states that there is some function $f$, such that for all integers $w,t>1$, every graph $G$ containing a wall of size $f(w,t)$, must contain either (i) a $K_t$-minor; or (ii) a small subset $A\subset V(G)$ of vertices, and a flat wall of size $w$ in $G\setminus A$. Kawarabayashi, Thomas and Wollan recently showed a self-contained proof of this theorem with the following two sets of parameters: (1) $f(w,t)=\Theta(t^{24}(t^2+w))$ with $|A|=O(t^{24})$, and (2) $f(w,t)=w^{2^{\Theta(t^{24})}}$ with $|A|\leq t-5$. The latter result gives the best possible bound on $|A|$. In this paper we improve their bounds to $f(w,t)=\Theta(t(t+w))$ with $|A|\leq t-5$. For the special case where the maximum vertex degree in $G$ is bounded by $D$, we show that, if $G$ contains a wall of size $\Omega(Dt(t+w))$, then either $G$ contains a $K_t$-minor, or there is a flat wall of size $w$ in $G$. This setting naturally arises in algorithms for the Edge-Disjoint Paths problem, with $D\leq 4$. Like the proof of Kawarabayashi et al., our proof is self-contained, except for using a well-known theorem on routing pairs of disjoint paths. We also provide efficient algorithms that return either a model of the $K_t$-minor, or a vertex set $A$ and a flat wall of size $w$ in $G\setminus A$. We complement our result for the low-degree scenario by proving an almost matching lower bound: namely, for all integers $w,t>1$, there is a graph $G$, containing a wall of size $\Omega(wt)$, such that the maximum vertex degree in $G$ is 5, and $G$ contains no flat wall of size $w$, and no $K_t$-minor.