On the dimension of posets with cover graphs of treewidth $2$

TL;DR

This paper proves that posets with cover graphs of treewidth 2 have bounded dimension, specifically at most 1276, extending known results from trees and outerplanar graphs to this boundary case.

Contribution

It establishes an upper bound on the dimension of posets with cover graphs of treewidth 2, filling a gap between known cases and demonstrating boundedness.

Findings

Posets with cover graphs of treewidth 2 have dimension at most 1276.

Bounded dimension results extend from trees and outerplanar graphs to treewidth 2.

The result supports the conjecture that such posets have bounded dimension.

Abstract

In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth $3$. In this paper we focus on the boundary case of treewidth $2$. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth $2$ (Bir\'o, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth $2$. We show that it is indeed the case: Every such poset has dimension at most $1276$.