三角剖分曲面中带洞球面的三角剖分 (Triangulated spheres with holes in triangulated surfaces)

Let $\mathbb{S}_h$ denote a sphere with $h$ holes. Given a triangulation $G$ of a surface $\mathbb{M}$, we consider the question of when $G$ contains a spanning subgraph $H$ such that $H$ is a triangulated $\mathbb{S}_h$. We give a new short proof of a theorem of Nevo and Tarabykin that every triangulation $G$ of the torus contains a spanning subgraph which is a triangulated cylinder. For arbitrary surfaces, we prove that every high facewidth triangulation of a surface with $h$ handles contains a spanning subgraph which is a triangulated $\mathbb{S}_{2h}$. We also prove that for every $0 \leq g' < g$ and $w \in \mathbb{N}$, there exists a triangulation of facewidth at least $w$ of a surface of Euler genus $g$ that does not have a spanning subgraph which is a triangulated $\mathbb{S}_{g'}$. Our results are motivated by, and have applications for, rigidity questions in the plane.

翻译：令 $\mathbb{S}_h$ 表示具有 $h$ 个洞的球面。给定曲面 $\mathbb{M}$ 的一个三角剖分 $G$，我们研究 $G$ 何时包含一个生成子图 $H$，使得 $H$ 是三角剖分的 $\mathbb{S}_h$。我们给出了 Nevo 和 Tarabykin 定理的一个新的简短证明：环面的每个三角剖分 $G$ 都包含一个作为三角剖分圆柱体的生成子图。对于任意曲面，我们证明了具有 $h$ 个环柄的曲面每个高面宽三角剖分都包含一个作为三角剖分 $\mathbb{S}_{2h}$ 的生成子图。我们还证明对于任意 $0 \leq g' < g$ 和 $w \in \mathbb{N}$，存在欧拉亏格为 $g$ 的曲面一个面宽至少为 $w$ 的三角剖分，其不包含作为三角剖分 $\mathbb{S}_{g'}$ 的生成子图。我们的研究动机源于平面刚性问题的应用需求，其结果对该领域具有应用价值。