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.

翻译：我们描述了一种高效变形球面图的有前景方法，扩展了Awartani与Henderson [Trans. AMS 1987] 以及Kobourov与Landis [JGAA 2006] 的早期研究。具体而言，我们提出了两种方法，通过将顶点沿经线移动至南半球来变形球面最短路径三角剖分；若存在此类变形，我们称该三角剖分为可下沉的。我们的第一种方法推广了Awartani与Henderson的经向壳层剥离构造：若三角剖分对偶图的特定定向是无环的，则该三角剖分可下沉。我们描述了一种简单的多项式时间算法，用于寻找给定球面三角剖分的可经向壳层剥离旋转（若存在）；同时构造了一个不存在可经向壳层剥离旋转的球面三角剖分。第二种方法基于可下沉性的线性规划表征。通过识别其最优基，我们证明该线性规划可在$O(n^{ω/2})$时间内求解，其中$ω$为矩阵乘法指数，前提是底层线性系统非奇异。除这些主要结果外，我们描述了将球面上3连通平面图的最短路径嵌入变形问题归约为三角剖分变形问题的方法，并提出一种高效算法以构造各中间边至多有一个弯曲的变形。最后，我们提出若干猜想并展示支持这些猜想的实验结果。