Remark on topological nature of upward planarity

TL;DR

This paper demonstrates that upward planarity in acyclic directed graphs is a topological property, showing that upward planar drawings are equivalent if connected by an orientation-preserving planar isotopy, thus confirming Selinger's conjecture.

Contribution

It combines graph theory and category theory to prove a topological invariance of upward planarity, providing a new proof of Selinger's conjecture.

Findings

Upward planar drawings are equivalent if connected by an orientation-preserving planar isotopy.

The result confirms that upward planarity is a topological property.

Provides a new proof of Selinger's conjecture using topological and categorical methods.

Abstract

The notion of an upward plane graph in graph theory and that of a progressive plane graph (or plane string diagram) in category theory are essentially the same thing. In this paper, we combine the ideas in graph theory and category theory to explain why and in what sense upward planarity is a topological property. The main result is that two upward planar drawings of an acyclic directed graph are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of $G$. This result gives a positive answer to Selinger's conjectue, whose strategy is different from the solution recently given by Delpeuch and Vicary.