Embedding Planar Graphs at Fixed Vertex Locations

@article{Pach2001EmbeddingPG,
  title={Embedding Planar Graphs at Fixed Vertex Locations},
  author={J{\'a}nos Pach and Rephael Wenger},
  journal={Graphs and Combinatorics},
  year={2001},
  volume={17},
  pages={717-728}
}
Abstract. Let G be a planar graph of n vertices, v1,…,vn, and let {p1,…,pn} be a set of n points in the plane. We present an algorithm for constructing in O(n2) time a planar embedding of G, where vertex vi is represented by point pi and each edge is represented by a polygonal curve with O(n) bends (internal vertices). This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly… 
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