Upward Point Set Embeddings of Paths and Trees
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We study upward planar straight-line embeddings (UPSE) of directed trees on given point sets. The given point set S has size at least the number of vertices in the tree. For the special case where the tree is a path P we show that: (a) If S is one-sided convex, the number of UPSEs equals the number of maximal monotone paths in P. (b) If S is in general position and P is composed by three maximal monotone paths, where the middle path is longer than the other two, then it always admits an UPSE on S. We show that the decision problem of whether there exists an UPSE of a directed tree with n vertices on a fixed point set S of n points is NP-complete, by relaxing the requirements of the previously known result which relied on the presence of cycles in the graph, but instead fixing position of a single vertex. Finally, by allowing extra points, we guarantee that each directed caterpillar on n vertices and with k switches in its backbone admits an UPSE on every set of n 2^k-2 points.
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