259 research outputs found

    â„“1-Rigid Graphs.

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    Cubic Partial Cubes from Simplicial Arrangements

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    We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure

    Dimension reduction by random hyperplane tessellations

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    Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map that sends x in K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R^n embeds into the Hamming cube {-1, 1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.Comment: 17 pages, 3 figures, minor update

    Graphs 4n4_n that are isometrically embeddable in hypercubes

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    A connected 3-valent plane graph, whose faces are qq- or 6-gons only, is called a {\em graph qnq_n}. We classify all graphs 4n4_n, which are isometric subgraphs of a mm-hypercube HmH_m.Comment: 18 pages, 25 drawing
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