Higher-rank trees arising from polyhedral graphs

TL;DR

This paper introduces a new family of higher-rank graphs inspired by graphical techniques, demonstrating their planarity and unique properties that distinguish them from 1-trees, with additional examples provided.

Contribution

The paper constructs a new class of higher-rank graphs based on polyhedral graphs, showing their planarity and unique properties, expanding the understanding of higher-rank trees.

Findings

Higher-rank graphs are planar $k$-trees for $2 \\le k \\le 4$

Higher-rank trees exhibit properties impossible for 1-trees

Additional examples of higher-rank planar trees outside the constructed family

Abstract

We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \cite{Lambek} and Johnstone \cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 \le k \le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties which are impossible for $1$-trees. Finally, we collect more examples of higher-rank planar trees which are not in our family.