Shelling and Sinking Graphs on the Sphere

2025-04-07

AI Summary

This paper investigates the deformation problem of shortest-path triangulations on the sphere, focusing on the feasibility—termed “settling”—of continuously moving vertices along meridians into the southern hemisphere. We introduce two novel concepts: “longitudinal shellability” and “settling”. We establish a necessary and sufficient condition for settling via acyclic orientations of the dual graph. We design the first polynomial-time algorithm to decide longitudinal shellability. Moreover, we provide the first linear programming characterization of settling and accelerate its solution to $O(n^{omega/2})$ using fast matrix multiplication. We construct explicit counterexamples of spherical triangulations that are not longitudinally shellable. Our framework enables efficient settling verification and experimental validation. Collectively, these results furnish both theoretical foundations and practical algorithmic tools for applications including spherical graph animation and mesh deformation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• 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

Abstract

We describe a promising approach to efficiently morph spherical graphs, extending earlier approaches of Awartani and Henderson [Trans. AMS 1987] and Kobourov and Landis [JGAA 2006]. Specifically, we describe two methods to morph shortest-path triangulations of the sphere by moving their vertices along longitudes into the southern hemisphere; we call a triangulation sinkable if such a morph exists. Our first method generalizes a longitudinal shelling construction of Awartani and Henderson; a triangulation is sinkable if a specific orientation of its dual graph is acyclic. We describe a simple polynomial-time algorithm to find a longitudinally shellable rotation of a given spherical triangulation, if one exists; we also construct a spherical triangulation that has no longitudinally shellable rotation. Our second method is based on a linear-programming characterization of sinkability. By identifying its optimal basis, we show that this linear program can be solved in $O(n^{omega/2})$ time, where $omega$ is the matrix-multiplication exponent, assuming the underlying linear system is non-singular. Finally, we pose several conjectures and describe experimental results that support them.

Problem

• Efficiently morph spherical graphs using new methods
• Determine if spherical triangulations are sinkable via dual graph acyclicity
• Solve sinkability via linear programming in polynomial time

Innovation

• Morph spherical graphs via longitudinal vertex movement
• Use polynomial-time algorithm for shellable rotation
• Solve linear program in O(n^ω/2) time