Treewidth is Polynomial in Maximum Degree on Weakly Sparse Graphs Excluding a Planar Induced Minor

A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ after vertex deletions and edge contractions. We show that for every $k$-vertex planar graph $H$, every graph $G$ excluding $H$ as an induced minor and $K_{t,t}$ as a subgraph has treewidth at most $\Delta(G)^{f(k,t)}$ where $\Delta(G)$ denotes the maximum degree of $G$. Without requiring the absence of a $K_{t,t}$ subgraph, Korhonen [JCTB '23] has shown the upper bound of $k^{O(1)} 2^{\Delta(G)^5}$ whose dependence in $\Delta(G)$ is exponential. Our result partially answers a question of Chudnovsky [Dagstuhl seminar '23] asking whether the treewidth of graphs with $\Delta(G)=O(\log{|V(G)|})$ excluding both a $k$-vertex planar graph as an induced minor and the biclique $K_{t,t}$ as a subgraph is in $O_{k,t}(\log |V(G)|)$. We confirm that the treewidth is in this case polylogarithmic in $|V(G)|$.