Treewidth is Polynomial in Maximum Degree on Weakly Sparse Graphs Excluding a Planar Induced Minor
Édouard Bonnet, Jędrzej Hodor, Tuukka Korhonen, Tomáš Masařík
TL;DR
The paper addresses bounding the treewidth of graphs that exclude a fixed planar induced minor and also forbid a biclique as a subgraph. It develops a suite of techniques—contraction–uncontraction to reduce maximum degree, star colorings with bounded color counts, and clustered edge-colorings—culminating in a product-structure approach via strong products to achieve a polynomial (in $ ext{Δ}(G)$) treewidth bound. The key result shows $ ext{tw}(G) leq ext{Δ}(G)^{f(k,t)}$ for appropriate $f$, and, under $ ext{Δ}(G)=O(\log|V(G)|)$, the treewidth is polylogarithmic in $|V(G)|$, yielding quasi-polynomial-time consequences for problems like Max Independent Set on these graph classes. Overall, the work advances the understanding of treewidth in weakly sparse graphs excluding planar induced minors and certain dense substructures, and it integrates product-structure ideas with refined sparsification and coloring techniques to obtain the bounds.
Abstract
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 $Δ(G)^{f(k,t)}$ where $Δ(G)$ 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 $k^{O(1)} 2^{Δ(G)^5}$ whose dependence in $Δ(G)$ is exponential. Our result partially answers a question of Chudnovsky [Dagstuhl seminar '23] asking whether the treewidth of graphs with $Δ(G)=O(\log{|V(G)|})$ 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 $|V(G)|$.