6,388 research outputs found

    3D Visibility Representations of 1-planar Graphs

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    We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    On Visibility Representations of Non-planar Graphs

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    A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge

    Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs

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    A graph is called a strong (resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a ϵ\epsilon-thick vertical line connecting them that intersects at most one other bar. We explore the relation among weak (resp. strong) bar 1-visibility graphs and other nearly planar graph classes. In particular, we study their relation to 1-planar graphs, which have a drawing with at most one crossing per edge; quasi-planar graphs, which have a drawing with no three mutually crossing edges; the squares of planar 1-flow networks, which are upward digraphs with in- or out-degree at most one. Our main results are that 1-planar graphs and the (undirected) squares of planar 1-flow networks are weak bar 1-visibility graphs and that these are quasi-planar graphs

    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

    L-Visibility Drawings of IC-planar Graphs

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    An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either a horizontal or vertical segment. As a byproduct of our drawing technique, we prove that an IC-plane graph has a RAC drawing in quadratic area with at most two bends per edge

    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

    A Quantitative Steinitz Theorem for Plane Triangulations

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    We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation GG with nn vertices can be embedded in R2\mathbb{R}^2 in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a 4n3×8n5×ζ(n)4n^3 \times 8n^5 \times \zeta(n) integer grid, where ζ(n)(500n8)τ(G)\zeta(n) \leq (500 n^8)^{\tau(G)} and τ(G)\tau(G) denotes the shedding diameter of GG, a quantity defined in the paper.Comment: 25 pages, 6 postscript figure
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