Minors and dimension

TL;DR

This paper proves that posets with bounded height and cover graphs excluding a fixed topological minor have bounded dimension, generalizing previous results and confirming a conjecture using advanced graph structure theorems.

Contribution

It establishes a broad bounded dimension result for posets with cover graphs excluding a fixed topological minor, extending prior special cases.

Findings

Bounded dimension for posets with cover graphs excluding a fixed topological minor.

Generalization of previous bounded dimension results.

Verification of a conjecture by Joret et al.

Abstract

It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.