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Monotone paths in planar convex subdivisions and polytopes
, 2012
"... Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most ∆ edges. Then, starting from every vertex there is a path with at least Ω(log ∆ n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivisi ..."
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Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most ∆ edges. Then, starting from every vertex there is a path with at least Ω(log ∆ n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivision of the plane into n convex faces where exactly k faces are unbounded. Then, there is a path with at least Ω(log(n/k) / log log(n/k)) edges that is monotone in some direction. This bound is also the best possible. Our methods are constructive and lead to efficient algorithms for computing monotone paths of lengths specified above. In 3space, we show that for every n ≥ 4, there exists a polytope P with n vertices, bounded vertex degrees, and triangular faces such that every monotone path on the 1skeleton of P has at most O(log 2 n) edges. We also construct a polytope Q with n vertices, and triangular faces, (with unbounded degree however), such that every monotone path on the 1skeleton of Q has at most O(log n) edges.
Long Monotone Paths on Simple 4Polytopes
, 2004
"... ... of vertices in a monotone path along edges of a ddimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a dpolytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, t ..."
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... of vertices in a monotone path along edges of a ddimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a dpolytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, the answer is “yes”, despite the fact that it is “no” if we restrict ourselves to the dualtocyclic polytopes. For each n ≥ 5, we exhibit a realization of a polartoneighborly 4dimensional polytope with n facets and a Hamilton path through its vertices that is monotone with respect to a linear objective function. This constrasts an earlier result, by which no polartoneighborly 6dimensional polytope with 9 facets admits a monotone Hamilton path.
Computing Disjoint Paths on Polytopes
, 2005
"... Abstract The HoltKlee Condition states that there exist at least d vertexdisjoint strictly monotone pathsfrom the source to the sink of a polytopal digraph consisting of the set of vertices and arcs of a ..."
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Abstract The HoltKlee Condition states that there exist at least d vertexdisjoint strictly monotone pathsfrom the source to the sink of a polytopal digraph consisting of the set of vertices and arcs of a