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Induced Minors and Coarse Tree Decompositions

Maria Chudnovsky, Julien Codsi, Ajaykrishnan E S, Daniel Lokshtanov

Abstract

Let $G$ be a graph, $S \subseteq V(G)$ be a vertex set in $G$ and $r$ be a positive integer. The distance $r$-independence number of $S$ is the size of the largest subset $I \subseteq S$ such that no pair $u$, $v$ of vertices in $I$ have a path on at most $r$ edges between them in $G$. It has been conjectured [Chudnovsky et al., arXiv, 2025] that for every positive integer $t$ there exist positive integers $c$, $d$ such that every graph $G$ that excludes both the complete bipartite graph $K_{t,t}$ and the grid $\boxplus_t$ as an induced minor has a tree decomposition in which every bag has (distance $1$) independence number at most $c(\log n)^d$. We prove a weaker version of this conjecture where every bag of the tree decomposition has distance $16(\log n + 1)$-independence number at most $c(\log n)^d$. On the way we also prove a version of the conjecture where every bag of the decomposition has distance $8$-independence number at most $2^{c (\log n)^{1-(1/d)}}$.

Induced Minors and Coarse Tree Decompositions

Abstract

Let be a graph, be a vertex set in and be a positive integer. The distance -independence number of is the size of the largest subset such that no pair , of vertices in have a path on at most edges between them in . It has been conjectured [Chudnovsky et al., arXiv, 2025] that for every positive integer there exist positive integers , such that every graph that excludes both the complete bipartite graph and the grid as an induced minor has a tree decomposition in which every bag has (distance ) independence number at most . We prove a weaker version of this conjecture where every bag of the tree decomposition has distance -independence number at most . On the way we also prove a version of the conjecture where every bag of the decomposition has distance -independence number at most .
Paper Structure (13 sections, 47 theorems, 46 equations)

This paper contains 13 sections, 47 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Proof Outline.
  3. Preliminaries
  4. Layered Family
  5. Rounding A--B Separator LP
  6. Using the Dual of A--B Separator LP
  7. Rounding Balanced Separator LP
  8. Using the Dual of Balanced Separator LP
  9. Some Results for an Auxiliary Class
  10. Vertex partitions of graphs with no $K_{t,t}$-induced-minors
  11. Proof of Theorem \ref{['thm:coarse_treewidth_our_class']}
  12. Proof of Theorem \ref{['thm:coarse_subpoly_treewidth_our_class']}
  13. Proof of Theorem \ref{['thm:coarse_induced_menger_our_class']}

Key Result

Theorem 1.1

Let $t$ be a positive integer and let $G$ be a $K_{t,t}$-induced-minor-free graph. Then there exists constants $c(t)$, $d(t)$ such that $G$ either has $\, \boxplus_t$ as an induced minor or has a tree decomposition $(T,\chi)$ such that for every $x\in V(T)$, the distance $16\log 2n$-independence num

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: hagerup1990guided; mitzenmacher2017probability
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • ...and 92 more