30,016 research outputs found

    Pixel and Voxel Representations of Graphs

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    We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for kk-outerplanar graphs with nn vertices, Θ(kn)\Theta(kn) pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, Θ(n2)\Theta(n^2) voxels are always sufficient and sometimes necessary for any nn-vertex graph. We improve this bound to Θ(nτ)\Theta(n\cdot \tau) for graphs of treewidth τ\tau and to O((g+1)2nlog2n)O((g+1)^2n\log^2n) for graphs of genus gg. In particular, planar graphs admit representations with O(nlog2n)O(n\log^2n) voxels

    Visibility Representations of Boxes in 2.5 Dimensions

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    We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane z=0z=0 and edges are unobstructed lines of sight parallel to the xx- or yy-axis. We prove that: (i)(i) Every complete bipartite graph admits a 2.5D-BR; (ii)(ii) The complete graph KnK_n admits a 2.5D-BR if and only if n19n \leq 19; (iii)(iii) Every graph with pathwidth at most 77 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an nn-vertex graph that admits a 2.5D-GBR has at most 4n6n4n - 6 \sqrt{n} edges and this bound is tight. Finally, we prove that deciding whether a given graph GG admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR Γ\Gamma is the set of bottom faces of the boxes in Γ\Gamma.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Crystal bases for quantum affine algebras and combinatorics of Young walls

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    In this paper, we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on the given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graphs for basic representations are characterized as the set of all reduced proper Young walls. The characters of basic representations can be computed easily by counting the number of colored blocks that have been added to the ground-state wall

    Situating graphs as workplace knowledge

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    We investigate the use and knowledge of graphs in the context of a large industrial factory. We are particularly interested in the question of "transparency", a question that has been extensively considered in the general literature on tool use, and more recently, by Michael Roth and his colleagues in the context of scientific work. Roth uses the notion of transparency to characterise instances of graph use by highly educated scientists in cases where the context was familiar: the scientists were able to read the situation "through" the graph. This paper explores the limits of the validity of the transparency metaphor. We present two vignettes of actual graph use by a factory worker, and contrast his actions and knowledge with that of a highly-qualified process engineer working on the same production line. We note that in neither case were the graphs transparent. We argue that a fuller account that describes a spectrum of transparency is needed, and we seek to achieve this by adopting some elements of a semiotic approach that enhance a strictly activity-theoretical view
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