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Improved Bounds for the Flat Wall Theorem

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) $f(w,t)=Θ(t^{24}(t^2+w))$ with $|A|=O(t^{24})$, and (2) $f(w,t)=w^{2^{Θ(t^{24})}}$ with $|A|\leq t-5$. The latter result gives the best possible bound on $|A|$. In this paper we improve their bounds to $f(w,t)=Θ(t(t+w))$ 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 $Ω(Dt(t+w))$, 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 $Ω(wt)$, such that the maximum vertex degree in $G$ is 5, and $G$ contains no flat wall of size $w$, and no $K_t$-minor.