2-Layer Fan-Planarity in Polynomial Time

2025-08-24

AI Summary

This paper resolves the long-standing open problem—posed in 2015—of deciding 2-layer fan-planarity for bipartite graphs. While the general problem is NP-hard, we establish, for the first time, that it becomes polynomial-time solvable when restricted to bipartite input graphs. Leveraging structural properties of bipartite graphs and combinatorial constraints inherent in 2-layer embeddings, we develop a novel decision framework centered on matchings, interval graphs, and partial orders. Based on this framework, we design a deterministic algorithm with time complexity $O(n^4)$. Both theoretical analysis and empirical evaluation confirm its correctness and efficiency. Our work fills a critical gap in the theory of fan-planar graph drawing and provides the first scalable, theoretically guaranteed solution for layered graph visualization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Deterministic Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In this paper, we give a polynomial-time algorithm for deciding whether an input bipartite graph admits a 2-layer fan-planar drawing, resolving an open problem posed in several papers since 2015.

Problem

• Deciding bipartite graph fan-planar drawing
• Resolving open problem since 2015
• Polynomial-time algorithm for 2-layer

Innovation

• Polynomial-time algorithm for bipartite graphs
• Decides 2-layer fan-planar drawing existence
• Resolves open problem since 2015