Shelling and Sinking Graphs on the Sphere

TL;DR

This work tackles efficient morphing of spherical graph embeddings by introducing sinkability and longitudinal shelling as core concepts. It develops two complementary approaches: a combinatorial shelling criterion with a polynomial-time rotation search and a linear-programming formulation that yields an $O(n^{ω/2})$ sinkability algorithm, along with reductions to planar morphs and a one-bend morph framework. The authors also extend morphing strategies to 3-connected embeddings and provide experimental evidence suggesting practical effectiveness, plus several conjectures guiding future research. Together, these results advance the understanding of spherical graph morphing and connect it to established planar morphing techniques, with potential to yield full morphs on the sphere under further developments.

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. In addition to these main results, we describe a reduction from morphing shortest-path embeddings of 3-connected planar graphs on the sphere to morphing triangulations, and we describe an efficient algorithm that constructs morphs where each intermediate edge has at most one bend. Finally, we pose several conjectures and describe experimental results that support them.

Figures (13)

• Figure 1.1: Three Schönhardt polyhedra, the corresponding spherical triangulations, and their stereographic projections into the plane. Depending on the twisting angle, these triangulations are (a) coherent and therefore longitudinally shellable, (b) sinkable but not longitudinally shellable, or (c) not sinkable.
• Figure 1.2: Four six-beaked shaddocks and stereographic projections of the corresponding spherical triangulations. Depending on the twisting angle, these triangulations are (a) coherent, (b) longitudinally shellable but not coherent, (c) sinkable but not longitudinally shellable (Jessen's icosahedron), or (d) not sinkable.
• Figure 2.1: An up-face $f_{\textsf{above}}(i)$, a down-face $f_{\textsf{below}}(j)$, and the north face $f_{\textsf{north}} = f_{\textsf{above}}(j)$ of a spherical triangulation. Edge $ij$ is a leg of both $f_{\textsf{above}}(i)$ and $f_{\textsf{below}}(j)$.
• Figure 2.2: Morphing a six-beaked shaddock triangulation j-oi-67d-sb-71 into a regular icosahedral triangulation; compare with Kobourov and Landis kl-mpgss-06.
• Figure 3.1: Stereographic projections of directed graphs ${\Gamma}^{\downarrow}$ (red straight edges), ${\Gamma}^{\rightarrow}$ (black curved edges), and ${\Gamma}^{\boldsymbol{\ssearrow\!\sswarrow}}$ (blue, second row) for the Platonic octahedral triangulation (which is longitudinally shellable) and a Schönhardt triangulation (which is not). Compare with Figure \ref{['F:schonhardt']}.

Theorems & Definitions (40)

• Lemma 2.1
• proof
• Corollary 2.2
• Corollary 2.3
• proof
• Conjecture 2.4
• Lemma 2.5
• proof : sketch
• Theorem 2.6
• proof