可平面图的离散迹定理与能量最小化弹簧嵌入
Discrete Trace Theorems and Energy Minimizing Spring Embeddings of Planar Graphs
摘要：
Tutte的弹簧嵌入定理指出，对于一个三连通的可平面图，如果图的外表面固定为平面中某个凸区域的补，并且所有其他顶点都位于其邻居的质心处，则这导致唯一嵌入，并且这个嵌入是平面的。它也相当快地遵循，这种嵌入最小化平方边长之和，条件是嵌入的外面。然而，如何嵌入该外表面根本不清楚。我们考虑嵌入这个外面的最小化问题，一些规范化，使平方边长的总和最小化。在这项工作中，我们展示了这个最优化问题和Schur补的图拉普拉斯算子的内部顶点之间的连接。我们证明了一些离散迹定理，并使用这些新的结果，显示谱等价的这舒尔补与边界拉普拉斯算子的一半权力的一大类图。利用这一结果，我们给出了这个最优化问题的理论保证，这促使一个算法来嵌入弹簧嵌入的外表面。
英文摘要：
Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to some normalization, so that the sum of squared edge lengths is minimized. In this work, we show the connection between this optimization problem and the Schur complement of the graph Laplacian with respect to the interior vertices. We prove a number of discrete trace theorems, and, using these new results, show the spectral equivalence of this Schur complement with the boundary Laplacian to the one-half power for a large class of graphs. Using this result, we give theoretical guarantees for this optimization problem, which motivates an algorithm to embed the outer face of a spring embedding.
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