Generalized Graph Drawing Algorithms Based on Transversal Structures and Separating Decompositions

Core Concepts

This article presents unified algorithms for straight-line and orthogonal drawing of planar maps with faces of degree at most 4, generalizing several classical algorithms based on transversal structures and separating decompositions.

Abstract

The article introduces a family of combinatorial structures called grand-Schnyder (GS) woods, which generalize the concepts of transversal structures and separating decompositions. The authors present:

A unified straight-line drawing algorithm for planar maps with faces of degree at most 4, based on face-counting and increasing functions on bipolar orientations associated to GS woods.

This algorithm generalizes the straight-line drawing algorithms of Fusy (for planar triangulations) and Barrière-Huemer (for simple quadrangulations).

A unified orthogonal drawing algorithm for the duals of planar maps with faces of degree at most 4, also based on face-counting and increasing functions.

This algorithm generalizes the orthogonal drawing algorithms of He (for the duals of planar triangulations) and Bernardi-Fusy (for the duals of simple quadrangulations).

The authors prove the correctness of the algorithms, analyze their time complexity and grid size, and discuss strategies to optimize the grid size.

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Key Insights Distilled From

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

by Oliv... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.18980.pdf

Deeper Inquiries

What are some potential applications of the generalized graph drawing algorithms presented in this article beyond planar maps

The generalized graph drawing algorithms presented in the article have potential applications beyond planar maps. One possible application is in network visualization, where complex networks can be represented and analyzed using these algorithms. This can be particularly useful in fields such as social network analysis, biological network modeling, and transportation network planning. The algorithms can also be applied in circuit design and optimization, where visualizing and analyzing complex circuit layouts can benefit from efficient graph drawing techniques. Additionally, these algorithms can be used in geographic information systems (GIS) for mapping and spatial analysis, as well as in software engineering for visualizing code dependencies and software architectures.

How could the drawing algorithms be extended to handle planar maps with faces of degree greater than 4

To extend the drawing algorithms to handle planar maps with faces of degree greater than 4, modifications and adaptations would be required. One approach could be to generalize the concept of 4-GS woods to accommodate higher-degree faces. This would involve redefining the combinatorial structures and algorithms to account for the increased complexity and connectivity of the maps. Additionally, the face-counting and increasing-function algorithms would need to be adjusted to handle the additional degrees of the faces while maintaining the efficiency and correctness of the drawing process.

Are there any other classical graph drawing algorithms that could be unified and generalized using the grand-Schnyder wood framework

The grand-Schnyder wood framework provides a versatile and systematic approach to graph drawing algorithms, making it possible to unify and generalize various classical algorithms. One such algorithm that could potentially be unified using the grand-Schnyder wood framework is the Tutte embedding algorithm. By incorporating the principles of 4-GS woods and their associated bipolar orientations, the Tutte embedding algorithm could be extended to handle a wider range of graph structures and provide more efficient and flexible graph drawings. This unified approach could lead to advancements in graph drawing theory and applications across different domains.