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Induced minors and subpolynomial treewidth

Maria Chudnovsky, Julien Codsi, David Fischer, Daniel Lokshtanov

TL;DR

The paper investigates induced-minor-free graph classes defined by forbidding K_{t,t} and W_{t×t} as induced minors under a clique-number bound and proves a subpolynomial bound on treewidth, specifically tw(G) ≤ 2^{c log^{1-ε} n} for some ε∈(0,1]. It introduces a novel barrier-based framework, including concepts of slim pairs, barriers, mineable pairs, and (F,r)-based graphs, to derive small separators and control treewidth. The approach combines a bipartite-structure lemma with barrier/mineability analysis and leverages existing star-coloring and minor theory (BHKM, TW16, TW17) to extend to the full class C_t^*. The results yield structural insights and potential algorithmic consequences for problems like MWIS on these graph families, highlighting subpolynomial growth in complexity tied to graph size.

Abstract

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-induced-minor-free if no induced minor of $G$ is isomorphic to a member of $\mathcal{H}$, We denote by $W_{t\times t}$ the $t$-by-$t$ hexagonal grid, and by $K_{t,t}$ the complete bipartite graph with both sides of the bipartition of size $t$. We show that the class of $\{K_{t,t},W_{t\times t}\}$-induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer $t$ there exist $ε\in (0,1]$ and $c \in \nat$ such that every $n$-vertex $\{K_{t,t},W_{t\times t}\}$-induced minor-free graph with no clique of size $t$ has treewidth at most $2^{c\log^{1-ε}n}$.

Induced minors and subpolynomial treewidth

TL;DR

The paper investigates induced-minor-free graph classes defined by forbidding K_{t,t} and W_{t×t} as induced minors under a clique-number bound and proves a subpolynomial bound on treewidth, specifically tw(G) ≤ 2^{c log^{1-ε} n} for some ε∈(0,1]. It introduces a novel barrier-based framework, including concepts of slim pairs, barriers, mineable pairs, and (F,r)-based graphs, to derive small separators and control treewidth. The approach combines a bipartite-structure lemma with barrier/mineability analysis and leverages existing star-coloring and minor theory (BHKM, TW16, TW17) to extend to the full class C_t^*. The results yield structural insights and potential algorithmic consequences for problems like MWIS on these graph families, highlighting subpolynomial growth in complexity tied to graph size.

Abstract

Given a family of graphs, we say that a graph is -induced-minor-free if no induced minor of is isomorphic to a member of , We denote by the -by- hexagonal grid, and by the complete bipartite graph with both sides of the bipartition of size . We show that the class of -induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer there exist and such that every -vertex -induced minor-free graph with no clique of size has treewidth at most .

Paper Structure

This paper contains 11 sections, 21 theorems, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation.
  3. Proof outline and organization
  4. The bipartite lemma
  5. Separating slim pairs in barred graphs
  6. From mines to barriers
  7. The class $\mathcal{C}_t$ is slim
  8. Separating slim pairs of vertices in $(F,r)$-based graphs in $\mathcal{C}_t$
  9. Bounding the treewidth of $(F,r)$-based graphs in $\mathcal{C}_t$
  10. Bounding the treewidth of $\mathcal{C}_t^*$
  11. Acknowledgments

Key Result

Theorem 1.1

For every $t \in \mathbb{N}$, there exist $\epsilon=\epsilon(t) \in (0,1]$, $c=c(t)\in {\mathbb N}$ and $d \in \mathbb{N}$ such that every $n$-vertex graph $G$ in $\mathcal{C}_t^*$ satisfies $tw(G) \leq 2^{a\log^{1-\epsilon} n }$.

Figures (4)

  • Figure 1: Outline of the proof of the existence of small separators for $q$-slim pairs in $(F,r)$-based graphs in $\mathcal{C}_t$.
  • Figure 2: Visualization of a barrier separating $u$ from $v$.
  • Figure 3: Visualization of the result of \ref{['lemma: mines to barriers']}.
  • Figure 4: Visualization of \ref{['thm: from many barrier one is good']}.

Theorems & Definitions (34)

  • Conjecture 1.1
  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 24 more