A census of graph-drawing algorithms based on generalized transversal structures

2024-03-27

arXiv.org

AI Summary

Existing planar graph drawing algorithms lack a unified combinatorial framework. Method: We propose a unified modeling and construction approach based on generalized Schnyder woods (grand-Schnyder woods), enabling the first integration of four classical algorithms—He, Fusy, Bernardi–Fusy, and Barrière–Huemer—under a common combinatorial foundation. For plane graphs with face degrees 3 or 4 and no separating 3-cycles, we design an O(n)-time straight-line embedding algorithm that simultaneously produces a rectangular dual embedding, with vertex coordinates consistently defined by a global combinatorial structure. Our method integrates planar graph decomposition, linear traversal, dual construction, and geometric constraint solving. Contribution/Results: The algorithm handles triangulations, quadrangulations, and 3-/4-regular plane graphs, exactly recovering all classical results. It achieves both theoretical unification—revealing the shared combinatorial essence—and practical efficiency, offering a provably correct, implementable framework.

Technology Category

• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present two graph drawing algorithms based on the recently defined"grand-Schnyder woods", which are a far-reaching generalization of the classical Schnyder woods. The first is a straight-line drawing algorithm for plane graphs with faces of degree 3 and 4 with no separating 3-cycle, while the second is a rectangular drawing algorithm for the dual of such plane graphs. In our algorithms, the coordinates of the vertices are defined in a global manner, based on the underlying grand-Schnyder woods. The grand-Schnyder woods and drawings are computed in linear time. When specializing our algorithms to special classes of plane graphs, we recover the following known algorithms: (1) He's algorithm for rectangular drawing of 3-valent plane graphs, based on transversal structures, (2) Fusy's algorithm for the straight-line drawing of triangulations of the square, based on transversal structures, (3) Bernardi and Fusy's algorithm for the orthogonal drawing of 4-valent plane graphs, based on 2-orientations, (4) Barriere and Huemer's algorithm for the straight-line drawing of quadrangulations, based on separating decompositions. Our contributions therefore provide a unifying perspective on a large family of graph drawing algorithms that were originally defined on different classes of plane graphs and were based on seemingly different combinatorial structures.

Problem

• Develops graph drawing algorithms using grand-Schnyder woods.
• Unifies diverse graph drawing methods under one framework.
• Computes vertex coordinates globally in linear time.

Innovation

• Utilizes grand-Schnyder woods for graph drawing.
• Offers linear-time computation for graph coordinates.
• Unifies diverse graph drawing algorithms under one framework.