Embedding a Planar Graph on a Given Point Set
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A point-set embedding of a planar graph G with n vertices on a set S of n points is a planar straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. We prove that the point-set embeddability problem is NP-complete for 3-connected planar graphs, answering a question of Cabello . We give an O(nlog^3n)-time algorithm for testing point-set embeddability of plane 3-trees, improving the algorithm of Moosa and Rahman . We prove that no set of 24 points can support all planar 3-trees with 24 vertices, partially answering a question of Kobourov . We compute 2-bend point-set embeddings of plane 3-trees in O(W^2) area, where W is the length of longest edge of the bounding box of S. Finally, we design algorithms for testing convex point-set embeddability of klee graphs and arbitrary planar graphs.