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 \(|V(G)|\).