
Compatible Paths on Labelled Point Sets
Let P and Q be finite point sets of the same cardinality in ℝ^2, each la...
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On Lshaped Point Set Embeddings of Trees: First Nonembeddable Examples
An Lshaped embedding of a tree in a point set is a planar drawing of th...
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A relaxation of the Directed Disjoint Paths problem: a global congestion metric helps
In the Directed Disjoint Paths problem, we are given a digraph D and a s...
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Duality pairs and homomorphisms to oriented and unoriented cycles
In the homomorphism order of digraphs, a duality pair is an ordered pair...
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Duality pairs and homomorphisms to oriented and unoriented cycles
In the homomorphism order of digraphs, a duality pair is an ordered pair...
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Growth of Random Trees by Leaf Attachment
We study the growth of a timeordered rooted tree by probabilistic attac...
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Recovering Trees with Convex Clustering
Convex clustering refers, for given {x_1, ..., x_n}⊂^p, to the minimizat...
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Upward Point Set Embeddings of Paths and Trees
We study upward planar straightline 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 onesided 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 NPcomplete, 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^k2 points.
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