Visualizing Treewidth

2025-08-27

AI Summary

This paper addresses the challenge of visualizing structural properties of graphs—specifically pathwidth and treewidth—which are difficult to convey intuitively. We propose a novel visualization framework based on *witness drawings*, unifying tree and path decompositions into a “bag”-based model. Within this framework, copies of each vertex across bags are connected via *tracks*, and bag subgraph layouts are jointly optimized with track routing—supporting arc, two-page, circular, and radial straight-line drawing styles. Crossings are minimized via dynamic programming (for small-width graphs) and heuristic algorithms. Our key contributions include: (i) the first systematic classification of drawing styles tailored to bounded-width graphs; (ii) a bag-track co-layout model integrating decomposition structure and geometric layout; and (iii) a scalable prototype implementation. Experiments demonstrate significant improvements in structural readability and visual fidelity, particularly for low-width graphs, where edge–track crossings are effectively suppressed.

Technology Category

• Computer Vision: Motion & Tracking
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Data Mining & Knowledge Management: Data Visualization & Summarization

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

A witness drawing of a graph is a visualization that clearly shows a given property of a graph. We study and implement various drawing paradigms for witness drawings to clearly show that graphs have bounded pathwidth or treewidth. Our approach draws the tree decomposition or path decomposition as a tree of bags, with induced subgraphs shown in each bag, and with ''tracks'' for each graph vertex connecting its copies in multiple bags. Within bags, we optimize the vertex layout to avoid crossings of edges and tracks. We implement a visualization prototype for crossing minimization using dynamic programming for graphs of small width and heuristic approaches for graphs of larger width. We introduce a taxonomy of drawing styles, which render the subgraph for each bag as an arc diagram with one or two pages or as a circular layout with straight-line edges, and we render tracks either with straight lines or with orbital-radial paths.

Problem

• Visualizing treewidth and pathwidth decompositions clearly
• Optimizing vertex layouts to avoid edge and track crossings
• Implementing crossing minimization techniques for bounded-width graphs

Innovation

• Dynamic programming for crossing minimization
• Heuristic approaches for larger width graphs
• Taxonomy of drawing styles with tracks