A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if we know, for each vertex, the number of neighbors in each of the four cardinal directions, the triangulation is not completely determined. In fact, we show that counting such triangulations is #P-hard via a reduction from #3-regular bipartite planar vertex cover.

翻译：平面上给定顶点集可能存在多种平面直线三角剖分，其中每个顶点的度数保持不变。这表明度数信息并不能完全确定三角剖分。我们证明，即使已知每个顶点在四个基本方向上的邻点数量，三角剖分仍不能被完全确定。事实上，通过从#3-正则二部平面顶点覆盖问题的归约，我们证明了此类三角剖分的计数属于#P难问题。