556 research outputs found

    Drawing 3-Polytopes with Good Vertex Resolution

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    Small grid embeddings of 3-polytopes

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    We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by O(27.55n)=O(188n)O(2^{7.55n})=O(188^{n}). If the graph contains a triangle we can bound the integer coordinates by O(24.82n)O(2^{4.82n}). If the graph contains a quadrilateral we can bound the integer coordinates by O(25.46n)O(2^{5.46n}). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

    Shortest Reconfiguration of Perfect Matchings via Alternating Cycles

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    Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar

    Embedding Stacked Polytopes on a Polynomial-Size Grid

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    A stacking operation adds a dd-simplex on top of a facet of a simplicial dd-polytope while maintaining the convexity of the polytope. A stacked dd-polytope is a polytope that is obtained from a dd-simplex and a series of stacking operations. We show that for a fixed dd every stacked dd-polytope with nn vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by O(n2log(2d))O(n^{2\log(2d)}), except for one axis, where the coordinates are bounded by O(n3log(2d))O(n^{3\log(2d)}). The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure

    Embedding Stacked Polytopes on a Polynomial-Size Grid

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    We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) x O(n4) x O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by G unter M. Ziegler, and is the rst nontrivial subexponential upper bound on the long-standing open question of the grid size necessary to embed arbitrary convex polyhedra, that is, about effcient versions of Steinitz's 1916 theorem. An immediate consequence of our result is that O(log n)-bit coordinates suffice for a greedy routing strategy in planar 3-trees.Deutsche Forschungsgemeinschaft (DFG) (Grant No. SCHU 2458/1-1

    Unital associahedra

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    We construct a topological cellular operad such that the algebras over its cellular chains are the homotopy unital A-infinity algebras of Fukaya-Oh-Ohta-Ono.Comment: 22 pages, EPS color figure
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