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Sparse Induced Subgraphs of Large Treewidth

Édouard Bonnet

TL;DR

The paper addresses sparsifying induced subgraphs while retaining large treewidth in graphs excluding $K_{t,t}$. It develops a framework combining grid-wall induced minors, clique-topological minors, and bounded-expansion principles to construct a 2-connected induced subgraph on $n$ vertices with treewidth at least $w$ and at most $(1+\varepsilon)n$ edges; the subgraph is either the line graph of a subdivision of a $w\times w$ wall or a spanning supergraph of a subdivision of a $K_{w,w}$. This extends Weissauer’s result on large bicliques or induced cycles and demonstrates near-optimal edge-density sparsification for induced subgraphs with large treewidth under a $K_{t,t}$-free constraint. The methods blend deep structural results from graph minors (Grohe–Marx, Korhonen, Fomin–Golovach–Thilikos) with probabilistic and Ramsey-style arguments to carefully reduce edge density while preserving large treewidth, yielding both theoretical and potential algorithmic consequences for induced-subgraph containment problems.

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.

Sparse Induced Subgraphs of Large Treewidth

TL;DR

The paper addresses sparsifying induced subgraphs while retaining large treewidth in graphs excluding . It develops a framework combining grid-wall induced minors, clique-topological minors, and bounded-expansion principles to construct a 2-connected induced subgraph on vertices with treewidth at least and at most edges; the subgraph is either the line graph of a subdivision of a wall or a spanning supergraph of a subdivision of a . This extends Weissauer’s result on large bicliques or induced cycles and demonstrates near-optimal edge-density sparsification for induced subgraphs with large treewidth under a -free constraint. The methods blend deep structural results from graph minors (Grohe–Marx, Korhonen, Fomin–Golovach–Thilikos) with probabilistic and Ramsey-style arguments to carefully reduce edge density while preserving large treewidth, yielding both theoretical and potential algorithmic consequences for induced-subgraph containment problems.

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 and , and real , there is an integer such that every graph with treewidth at least and no subgraph admits a 2-connected -vertex induced subgraph with treewidth at least and at most 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.
Paper Structure (14 sections, 16 theorems, 1 equation, 1 figure, 1 table)

This paper contains 14 sections, 16 theorems, 1 equation, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subgraphs, induced subgraphs, neighborhoods
  4. Tree-decompositions, adhesions, brambles
  5. Minors, induced minors, branch sets
  6. Subdivisions, quasi-subdivisions, topological minors
  7. Degeneracy and edge density
  8. Shallow minors and expansion
  9. Grids and walls
  10. Preparatory lemmas: grid and wall induced minors
  11. Finding a large clique as a topological minor
  12. Upper bound on edge density, lower bound on subdivision length
  13. Decreasing the density of extra edges in the topological minor
  14. Wrapping up

Key Result

Theorem 1

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

Figures (1)

  • Figure 1: The $5 \times 5$ grid (left) and the $5 \times 5$ wall (right).

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Theorem 8: Korhonen23, Theorem 1
  • Theorem 9: FominGT11,Theorem 2
  • Theorem 10: GroheM15, weaker statement than Theorem 4.1
  • ...and 6 more