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Shelling and Sinking Graphs on the Sphere

Authors Jeff Erickson , Christian Howard

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  • Theory of computation → Computational geometry
Keywords
  • morphing
  • planar graphs
  • spherical graph drawing
  • longitudinal shelling

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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^{ω/2}) time, where ω 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.

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Jeff Erickson and Christian Howard. Shelling and Sinking Graphs on the Sphere. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.47

Author Details

Jeff Erickson
  • University of Illinois Urbana-Champaign, IL, USA
Christian Howard
  • University of Illinois Urbana-Champaign, IL, USA

Acknowledgements

The authors thank Eli Kujawa for early helpful discussions and the anonymous reviewers for their helpful comments and suggestions.

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