3,045 research outputs found

    Planar and Poly-Arc Lombardi Drawings

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    In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.Comment: Expanded version of paper appearing in the 19th International Symposium on Graph Drawing (GD 2011). 16 pages, 8 figure

    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

    Drawing Planar Graphs with Few Geometric Primitives

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    We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let nn denote the number of vertices of a graph. We show that trees can be drawn with 3n/43n/4 straight-line segments on a polynomial grid, and with n/2n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n−17)/3(8n-17)/3 segments on an O(n)×O(n2)O(n)\times O(n^2) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/23n/2 edges on an O(n)×O(n2)O(n)\times O(n^2) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n−11)/3(5n - 11)/3 arcs. This is significantly smaller than the lower bound of 2n2n for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017

    Universal Point Sets for Drawing Planar Graphs with Circular Arcs

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    Drawing Graphs with Circular Arcs and Right-Angle Crossings

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    In a RAC drawing of a graph, vertices are represented by points in the plane, adjacent vertices are connected by line segments, and crossings must form right angles. Graphs that admit such drawings are RAC graphs. RAC graphs are beyond-planar graphs and have been studied extensively. In particular, it is known that a RAC graph with n vertices has at most 4n - 10 edges. We introduce a superclass of RAC graphs, which we call arc-RAC graphs. A graph is arc-RAC if it admits a drawing where edges are represented by circular arcs and crossings form right angles. We provide a Tur\'an-type result showing that an arc-RAC graph with n vertices has at most 14n - 12 edges and that there are n-vertex arc-RAC graphs with 4.5n - o(n) edges

    Lombardi Drawings of Graphs

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    We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

    Achieving Good Angular Resolution in 3D Arc Diagrams

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    We study a three-dimensional analogue to the well-known graph visualization approach known as arc diagrams. We provide several algorithms that achieve good angular resolution for 3D arc diagrams, even for cases when the arcs must project to a given 2D straight-line drawing of the input graph. Our methods make use of various graph coloring algorithms, including an algorithm for a new coloring problem, which we call localized edge coloring.Comment: 12 pages, 5 figures; to appear at the 21st International Symposium on Graph Drawing (GD 2013

    Drawing Trees with Perfect Angular Resolution and Polynomial Area

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    We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

    Obstacle Numbers of Planar Graphs

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    Given finitely many connected polygonal obstacles O1,…,OkO_1,\dots,O_k in the plane and a set PP of points in general position and not in any obstacle, the {\em visibility graph} of PP with obstacles O1,…,OkO_1,\dots,O_k is the (geometric) graph with vertex set PP, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph GG is the smallest integer kk such that GG is the visibility graph of a set of points with kk obstacles. If GG is planar, we define the planar obstacle number of GG by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of GG). In this paper, we prove that the maximum planar obstacle number of a planar graph of order nn is n−3n-3, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least 33 is 11.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017
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