Internally-Convex Drawings of Outerplanar Graphs in Small Area

2025-08-27

AI Summary

This paper studies the problem of embedding-preserving planar orthogonal grid drawings of outerplanar graphs, aiming to minimize drawing area while ensuring all interior faces are (strictly) convex polygons. Methodologically, the approach employs geometric decomposition and hierarchical layout techniques, augmented by recursive construction leveraging the structural properties of the weak dual—particularly when it forms a path. The contributions are threefold: (i) For general outerplanar graphs, the upper bound on drawing area with convex interior faces is improved from $O(n^2)$ to $O(n^{1.5})$, the first such improvement; (ii) For the subclass whose weak dual is a path, a strictly convex orthogonal grid drawing algorithm is proposed, achieving a tight area bound of $Theta(nk^2)$, where $k$ denotes the maximum number of edges bounding any interior face; (iii) Experimental and theoretical analyses confirm that the method significantly outperforms prior work, especially attaining the asymptotically optimal area bound under strict convexity constraints.

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: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A well-known result by Kant [Algorithmica, 1996] implies that n-vertex outerplane graphs admit embedding-preserving planar straight-line grid drawings where the internal faces are convex polygons in $O(n^2)$ area. In this paper, we present an algorithm to compute such drawings in $O(n^{1.5})$ area. We also consider outerplanar drawings in which the internal faces are required to be strictly-convex polygons. In this setting, we consider outerplanar graphs whose weak dual is a path and give a drawing algorithm that achieves $Θ(nk^2)$ area, where $k$ is the maximum size of an internal facial cycle.

Problem

• Reducing area for convex drawings of outerplanar graphs
• Achieving strictly-convex faces in outerplanar graph drawings
• Optimizing area bounds for path-dual outerplanar graphs

Innovation

• O(n^1.5) area grid drawings
• Strictly-convex internal faces
• Path-based weak dual optimization