A composition theory for upward planar orders

TL;DR

This paper develops a composition theory for upward planar orders on acyclic directed graphs, enabling the calculation and understanding of these orders in the context of graphical calculus for monoidal categories.

Contribution

It introduces a composition framework for upward planar orders, showing that their composition preserves the upward planar order property.

Findings

Composition of upward planar orders results in an upward planar order.

Provides a practical method for calculating upward planar orders.

Enhances the combinatorial understanding of upward plane graphs.

Abstract

An upward planar order on an acyclic directed graph $G$ is a special linear extension of the edge poset of $G$ that satisfies the nesting condition. This order was introduced to combinatorially characterize upward plane graphs and progressive plane graphs (commonly known as plane string diagrams). In this paper, motivated by the theory of graphical calculus for monoidal categories, we establish a composition theory for upward planar orders. The main result is that the composition of upward planar orders is an upward planar order. This theory provides a practical method to calculate the upward planar order of a progressive plane graph or an upward plane graph.