Shelling and Sinking Graphs on the Sphere

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.