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Grid Minors and Products

Vida Dujmović, Pat Morin, David R. Wood, David Worley

TL;DR

This paper analyzes grid minors in graph products, linking them to treewidth via the grid minor number $\mathrm{gm}(G)$. It proves a general lower bound $\mathrm{gm}(G_1\Box G_2) \in \Omega(\sqrt{n})$ for two connected $n$-vertex graphs, and shows tightness for products with a star and an $n$-vertex tree, where all product variants have $\mathrm{gm} = O(\sqrt{n})$. The authors also provide upper bounds, establishing exact constants for $\mathrm{gm}$ in products of stars and trees: $c=2$ for Cartesian, $c=3$ for lexicographic, and a range $2.5 \le c \le 3$ for strong products, with precise constructions and reductions using subdivided stars and disjoint vertical paths. These results place graph products in a much stronger grid-minor regime than general graphs and connect to the Planar Graph Product Structure Theory. The work also outlines open questions about tightening constants and preserving treewidth in product decompositions of planar graphs.

Abstract

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $Ω(\sqrt{n})\timesΩ(\sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $ω(\sqrt{n})\timesω(\sqrt{n})$ grid minor.

Grid Minors and Products

TL;DR

This paper analyzes grid minors in graph products, linking them to treewidth via the grid minor number . It proves a general lower bound for two connected -vertex graphs, and shows tightness for products with a star and an -vertex tree, where all product variants have . The authors also provide upper bounds, establishing exact constants for in products of stars and trees: for Cartesian, for lexicographic, and a range for strong products, with precise constructions and reductions using subdivided stars and disjoint vertical paths. These results place graph products in a much stronger grid-minor regime than general graphs and connect to the Planar Graph Product Structure Theory. The work also outlines open questions about tightening constants and preserving treewidth in product decompositions of planar graphs.

Abstract

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected -vertex graphs contains an grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any -vertex tree has no grid minor.
Paper Structure (9 sections, 16 theorems, 30 equations, 7 figures)

This paper contains 9 sections, 16 theorems, 30 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. The Lower Bound
  5. Connected Graphs Contain Large Subdivided Stars
  6. The Product of Two Special Trees
  7. Upper Bound
  8. Product of Stars and Trees
  9. Open Problems

Key Result

Lemma 2

For any positive integer $n$ and any $n$-vertex connected graph $G$, $K_{n} \preceq G\mathbin{\Box} S_n$.

Figures (7)

  • Figure 1: The products of two paths and of a star and a path.
  • Figure 2: Finding paths in the tree $T'$ induced by vertices of height at least $p=4$.
  • Figure 3: Visualizing the product in \ref{['star_times_star']}
  • Figure 4: The $sp\times sp$ grid can be partitioned into $(sp/p)^2 = s^2$ subgrids, each of which is a $p\times p$ grid. (The case $sp=12$ and $p=4$ is shown here.)
  • Figure 5: One of the $p\times p$ subgrids used in the proof of \ref{['star_times_star']}.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 21 more