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Product structure of graphs with an excluded minor

Freddie Illingworth, Alex Scott, David R. Wood

Abstract

This paper shows that $K_t$-minor-free (and $K_{s, t}$-minor-free) graphs $G$ are subgraphs of products of a tree-like graph $H$ (of bounded treewidth) and a complete graph $K_m$. Our results include optimal bounds on the treewidth of $H$ and optimal bounds (to within a constant factor) on $m$ in terms of the number of vertices of $G$ and the treewidth of $G$. These results follow from a more general theorem whose corollaries include a strengthening of the celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc. 1990] and the Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM 2020].

Product structure of graphs with an excluded minor

Abstract

This paper shows that -minor-free (and -minor-free) graphs are subgraphs of products of a tree-like graph (of bounded treewidth) and a complete graph . Our results include optimal bounds on the treewidth of and optimal bounds (to within a constant factor) on in terms of the number of vertices of and the treewidth of . These results follow from a more general theorem whose corollaries include a strengthening of the celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc. 1990] and the Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM 2020].

Paper Structure

This paper contains 10 sections, 28 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tree-decompositions and treewidth
  4. Partitions
  5. Hitting sets
  6. Main theorem and O(tw(G) blow-up
  7. Layered treewidth: planar and apex-minor-free graphs
  8. Blow-up O(rootn)
  9. Concluding Remarks
  10. Simple Treewidth

Key Result

Theorem 1

Every $n$-vertex $K_t$-minor-free graph $G$ has treewidth $\mathop{\mathrm{tw}}\nolimits(G)< t^{3/2} n^{1/2}$.

Theorems & Definitions (37)

  • Theorem 1: AST90
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7: DJMMUW20
  • Lemma 9
  • proof
  • Lemma 10: AST90
  • ...and 27 more