  A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained
from $G$ by vertex deletions and edge contractions. We show that for every
$k$-vertex planar graph $H$, every graph $G$ excluding $H$ as an induced minor
has treewidth at most $\Delta(G)^{2^{O(k)}}$ where $\Delta(G)$ denotes the
maximum degree of $G$. Previously, Korhonen [JCTB '23] has shown the upper
bound of $k^{O(1)} 2^{\Delta(G)^5}$ whose dependence in $\Delta(G)$ is
exponential. More precisely, we show that every graph $G$ excluding as induced
minors a $k$-vertex planar graph and a $q$-vertex graph has treewidth at most
$k^{O(1)} \cdot \Delta(G)^{f(q)}$ with $f(q) = 2^{O(q)}$. A direct consequence
of our result is that for every hereditary graph class $\mathcal C$, if graphs
of $\mathcal C$ have treewidth bounded by a function of their maximum degree,
then they in fact have treewidth polynomial in their maximum degree.

Combinatorics;Discrete Mathematics;Data Structures and Algorithms