2,764 research outputs found
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Separation of line drawings based on split faces for 3D object reconstruction
© 2014 IEEE. Reconstructing 3D objects from single line drawings is often desirable in computer vision and graphics applications. If the line drawing of a complex 3D object is decomposed into primitives of simple shape, the object can be easily reconstructed. We propose an effective method to conduct the line drawing separation and turn a complex line drawing into parametric 3D models. This is achieved by recursively separating the line drawing using two types of split faces. Our experiments show that the proposed separation method can generate more basic and simple line drawings, and its combination with the example-based reconstruction can robustly recover wider range of complex parametric 3D objects than previous methods.This work was supported by grants from Science, Industry, Trade, and Information Technology Commission of Shenzhen Municipality (No. JC201005270378A),
Guangdong Innovative Research Team Program
(No. 201001D0104648280), Shenzhen Basic Research Program (JCYJ20120617114614438, JC201005270350A, JCYJ20120903092050890), Scientific Research Fund of
Hunan Provincial Education Department (No. 13C073), Industrial Technology Research and Development Program of Hengyang Science and Technology Bureau (No.2013KG75), and the Construct Program of the Key Discipline in Hunan Provinc
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
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