Sparse Induced Subgraphs of Large Treewidth

TL;DR

This paper proves that large treewidth graphs without certain bipartite subgraphs contain large induced subgraphs with controlled edge count, extending known results about large bicliques and cycles.

Contribution

It establishes the existence of large induced subgraphs with high treewidth and bounded edges in graphs with large treewidth and no K_{t,t} subgraph, generalizing previous results.

Findings

Large induced subgraphs with treewidth at least w exist in graphs with sufficiently large treewidth.

Such subgraphs can be either subdivided walls, line graphs, or spanning supergraphs of subdivided bicliques.

The results extend prior work on large bicliques or cycles in graphs of large treewidth.

Abstract

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.