3,518 research outputs found

    Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

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    We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes (beyond-planar graphs is a collective term for the primary attempts to generalize the planar graphs) to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every nn-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n)O(n) \times O(n). Then, we show that every nn-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n3)×O(n3)O(n^3) \times O(n^3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line

    Recognizing and Drawing IC-planar Graphs

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    IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph GG with nn vertices, we present an O(n)O(n)-time algorithm that computes a straight-line drawing of GG in quadratic area, and an O(n3)O(n^3)-time algorithm that computes a straight-line drawing of GG with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

    Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

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    We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing, Bordeaux, 201

    Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

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    We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) 33-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the 33-connected case) to this setting. Precisely, for any cylindric essentially internally 33-connected map GG with nn vertices, we can obtain in linear time a periodic (in xx) straight-line drawing of GG that is crossing-free and internally (weakly) convex, on a regular grid Z/wZ×[0..h]\mathbb{Z}/w\mathbb{Z}\times[0..h], with w2nw\leq 2n and hn(2d+1)h\leq n(2d+1), where dd is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially 33-connected map GG on the torus (i.e., 33-connected in the periodic representation) with nn vertices, we can compute in linear time a periodic straight-line drawing of GG that is crossing-free and (weakly) convex, on a periodic regular grid Z/wZ×Z/hZ\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}, with w2nw\leq 2n and h1+2n(c+1)h\leq 1+2n(c+1), where cc is the face-width of GG. Since c2nc\leq\sqrt{2n}, the grid area is O(n5/2)O(n^{5/2}).Comment: 37 page

    Snapping Graph Drawings to the Grid Optimally

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    In geographic information systems and in the production of digital maps for small devices with restricted computational resources one often wants to round coordinates to a rougher grid. This removes unnecessary detail and reduces space consumption as well as computation time. This process is called snapping to the grid and has been investigated thoroughly from a computational-geometry perspective. In this paper we investigate the same problem for given drawings of planar graphs under the restriction that their combinatorial embedding must be kept and edges are drawn straight-line. We show that the problem is NP-hard for several objectives and provide an integer linear programming formulation. Given a plane graph G and a positive integer w, our ILP can also be used to draw G straight-line on a grid of width w and minimum height (if possible).Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Monotone Grid Drawings of Planar Graphs

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    A monotone drawing of a planar graph GG is a planar straight-line drawing of GG where a monotone path exists between every pair of vertices of GG in some direction. Recently monotone drawings of planar graphs have been proposed as a new standard for visualizing graphs. A monotone drawing of a planar graph is a monotone grid drawing if every vertex in the drawing is drawn on a grid point. In this paper we study monotone grid drawings of planar graphs in a variable embedding setting. We show that every connected planar graph of nn vertices has a monotone grid drawing on a grid of size O(n)×O(n2)O(n)\times O(n^2), and such a drawing can be found in O(n) time

    On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

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    We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017
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