Sparse Induced Subgraphs of Large Treewidth

Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour [JCTB '86] or by a classic result of Chekuri and Chuzhoy [SODA '15], we show that for any natural numbers $t$ and $w$, and real $\varepsilon > 0$, there is an integer $W := W(t,w,\varepsilon)$ such that every graph with treewidth at least $W$ and no $K_{t,t}$ subgraph admits a 2-connected $n$-vertex induced subgraph with treewidth at least $w$ and at most $(1+\varepsilon)n$ edges. The induced subgraph is either a subdivided wall, or its line graph, or a spanning supergraph of a subdivided biclique. This in particular extends a result of Weissauer [JCTB '19] that graphs of large treewidth have a large biclique subgraph or a long induced cycle.