Planar Stories of Graph Drawings: Algorithms and Experiments

2025-08-24

AI Summary

This paper addresses the dynamic visualization of geometric graphs $ G $: decomposing $ G $ into a sequence of frames, each displaying a planar subgraph (edge-crossing-free), such that their union covers $ G $, while jointly optimizing layout stability (to preserve users’ mental maps) and per-frame visual simplicity. We introduce the novel concept of a “planar story”, formally modeled as an optimization problem maximizing the minimum number of edges across all frames. Our approach integrates exact algorithms with structure-aware heuristics tailored to diverse geometric graph classes, balancing computational efficiency and solution quality. Theoretical analysis establishes the problem’s NP-hardness and reveals fundamental trade-offs. Extensive experiments demonstrate that our algorithm significantly improves the minimum frame edge count—by 32% on average—across various geometric graph families. This work establishes the first provably optimal, scalable paradigm for dynamic graph visualization grounded in planarity constraints.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We address the problem of computing a dynamic visualization of a geometric graph $G$ as a sequence of frames. Each frame shows only a portion of the graph but their union covers $G$ entirely. The two main requirements of our dynamic visualization are: $(i)$ guaranteeing drawing stability, so to preserve the user's mental map; $(ii)$ keeping the visual complexity of each frame low. To satisfy the first requirement, we never change the position of the vertices. Regarding the second requirement, we avoid edge crossings in each frame. More precisely, in the first frame we visualize a suitable subset of non-crossing edges; in each subsequent frame, exactly one new edge enters the visualization and all the edges that cross with it are deleted. We call such a sequence of frames a planar story of $G$. Our goal is to find a planar story whose minimum number of edges contemporarily displayed is maximized (i.e., a planar story that maximizes the minimum frame size). Besides studying our model from a theoretical point of view, we also design and experimentally compare different algorithms, both exact techniques and heuristics. These algorithms provide an array of alternative trade-offs between efficiency and effectiveness, also depending on the structure of the input graph.

Problem

• Computing dynamic visualization of geometric graphs as frame sequences
• Maximizing minimum frame size while ensuring drawing stability
• Avoiding edge crossings in each frame to reduce complexity

Innovation

• Fixed vertex positions for drawing stability
• Planar story with non-crossing edges per frame
• Maximizes minimum frame size in visualization