Algebraically grid-like graphs have large tree-width

TL;DR

This paper establishes that graphs algebraically resembling grids, characterized by specific cycle sums and metric distortions, necessarily have large tree-width, linking algebraic properties to structural graph complexity.

Contribution

It introduces a new algebraic framework for identifying grid-like structures in graphs and proves that such structures imply large tree-width, extending the understanding of graph minors.

Findings

Graphs with algebraic grid-like properties have large tree-width.

Cycle sums and metric distortion bounds characterize grid-like graphs.

Large grid minors are guaranteed by algebraic cycle conditions.

Abstract

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long cycle on the perimeter, which is the $\mathbb{F}_2$-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width: Let $k, p$ be integers, $\gamma$ a real number and $G$ a graph. Suppose that $G$ contains a cycle of length at least $2 \gamma p k$ which is the $\mathbb{F}_2$-sum of cycles of length at most $p$ and whose metric is distorted by a factor of at most $\gamma$. Then $G$ has tree-width at least $k$.