A composition theory for upward planar orders

2025-05-20

AI Summary

Upward planar extensions of directed acyclic graphs (DAGs) lack a compositional theory, hindering structured synthesis in graphical calculi. Method: We integrate poset theory, topological planar graph embeddings, and categorical diagrammatic reasoning to develop a compositional linear extension framework for single-source/single-sink DAGs, introducing a nested-constraint-based linear extension construction technique. Contributions: (1) We establish the closure of upward planar orders under order composition—the first proof of compositional closure for this class; (2) we provide a polynomial-time algorithm for constructing upward planar orders of asymptotically planar DAGs; (3) our framework enables automated normalization and equivalence checking of string diagrams in monoidal categories, thereby filling a foundational gap in the theory of structured composition within diagrammatic calculi. This work bridges order theory, planarity, and categorical semantics, yielding the first compositional linear extension theory for upward planar DAGs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Characterize upward planar orders combinatorially
• Develop composition theory for upward planar orders
• Calculate upward planar orders for specific graphs

Innovation

• Composition theory for upward planar orders
• Linear extension satisfying nesting condition
• Practical calculation method for graphs