A composition theory for upward planar orders
Xue Dong, Xuexing Lu, Yu Ye
TL;DR
The paper addresses how to compute upward planar orders for upward plane and progressive plane graphs by introducing a composition theory that mirrors layer-based diagram calculations in graphical calculus. It defines admissible UPO-graphs with boundary-aware conditions and a shuffle-style composition rule, proving associativity and that composition preserves admissibility. The main theorem shows that the class of admissible UPO-graphs is closed under composition, enabling modular, layer-wise construction of upward planar orders for complex diagrams. This framework provides a practical and conceptually clear method to derive upward planar orders for progressive plane graphs and upward plane graphs, connecting graph drawing with monoidal-category graphical calculus.
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.