537 research outputs found

    Embedding a graph in the grid of a surface with the minimum number of bends is NP-hard

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    This paper is devoted to the study of graph embeddings in the grid of non-planar surfaces. We provide an adequate model for those embeddings and we study the complexity of minimizing the number of bends. In particular, we prove that testing whether a graph admits a rectilinear (without bends) embedding essentially equivalent to a given embedding, and that given a graph, testing if there exists a surface such that the graph admits a rectilinear embedding in that surface are NP-complete problems and hence the corresponding optimization problems are NP-hard

    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

    Single bend wiring on surfaces

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    The following problem of rectilinear routing is studied: given pairs of points on a surface and a set of permissible orthogonal paths joining them, whether is it possible to choose a path for each pair avoiding all intersections. We prove that if each pair has one or two possible paths to join it, then the problem is solvable in quadratic time, and otherwise it is NP-complete. From that result, we will obtain that the problem of finding a surface of minimum genus on which the wires can be laid out with only one bend is NP-hard

    New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity

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    Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity
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