8,312 research outputs found

    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

    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

    On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs

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    In this paper we study the area requirements of planar greedy drawings of triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family H\cal H of subdivisions of triconnected plane graphs and claimed that every planar greedy drawing of the graphs in H\mathcal H respecting the prescribed plane embedding requires exponential area. However, we show that every nn-vertex graph in H\cal H actually has a planar greedy drawing respecting the prescribed plane embedding on an O(n)×O(n)O(n)\times O(n) grid. This reopens the question whether triconnected planar graphs admit planar greedy drawings on a polynomial-size grid. Further, we provide evidence for a positive answer to the above question by proving that every nn-vertex Halin graph admits a planar greedy drawing on an O(n)×O(n)O(n)\times O(n) grid. Both such results are obtained by actually constructing drawings that are convex and angle-monotone. Finally, we consider α\alpha-Schnyder drawings, which are angle-monotone and hence greedy if α30\alpha\leq 30^\circ, and show that there exist planar triangulations for which every α\alpha-Schnyder drawing with a fixed α<60\alpha<60^\circ requires exponential area for any resolution rule

    A Universal Point Set for 2-Outerplanar Graphs

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    A point set SR2S \subseteq \mathbb{R}^2 is universal for a class G\cal G if every graph of G{\cal G} has a planar straight-line embedding on SS. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a sub-quadratic universal point set for them is one of the most fascinating open problems in Graph Drawing. Motivated by the fact that outerplanarity is a key property for the existence of small universal point sets, we study 2-outerplanar graphs and provide for them a universal point set of size O(nlogn)O(n \log n).Comment: 23 pages, 11 figures, conference version at GD 201

    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

    The Complexity of Simultaneous Geometric Graph Embedding

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    Given a collection of planar graphs G1,,GkG_1,\dots,G_k on the same set VV of nn vertices, the simultaneous geometric embedding (with mapping) problem, or simply kk-SGE, is to find a set PP of nn points in the plane and a bijection ϕ:VP\phi: V \to P such that the induced straight-line drawings of G1,,GkG_1,\dots,G_k under ϕ\phi are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for R\exists\mathbb{R}, the existential theory of the reals. Hence the problem kk-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to kk-SGE, with the property that both numbers kk and nn are linear in the number of pseudolines. This implies not only the R\exists\mathbb{R}-hardness result, but also a 22Ω(n)2^{2^{\Omega (n)}} lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only 22Ω(n)2^{2^{\Omega (\sqrt{n})}}

    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
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