3-多面体的小网格嵌入
Small grid embeddings of 3-polytopes
摘要：
我们介绍了一个算法，嵌入一个给定的3-连通平面图作为一个凸3-多面体与整数坐标。坐标的大小以$O(2^{7.55n})=O(188^{n})$为界。如果图形包含三角形，我们可以通过$O(2^{4.82n})$来限制整数坐标。如果图包含四边形，我们可以通过$O(2^{5.46n})$来限制整数坐标。该算法的关键部分是找到一个凸平面嵌入，其边缘可以被加权，使得加权边缘之和（视为向量）在每个点处抵消。众所周知，这可以通过应用Tutte技术来保证内部顶点。我们将展示如何扩展Tutte的想法，以构建一个平面嵌入的加权向量和取消也对边界面的顶点。
英文摘要：
We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
