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Confluent drawings: Visualizing NonPlanar Diagrams in a Planar Way
 GRAPH DRAWING (PROC. GD ’03), VOLUME 2912 OF LECTURE NOTES COMPUT. SCI
, 2003
"... We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing nonplanar graphs in a planar way. This approach allows us to draw, in a crossingfree manner, graphs—such as software interaction diagrams—that would normally have many cro ..."
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Cited by 28 (7 self)
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We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing nonplanar graphs in a planar way. This approach allows us to draw, in a crossingfree manner, graphs—such as software interaction diagrams—that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as “tracks” (similar to train tracks). Producing such confluent drawings automatically from a graph with many crossings is quite challenging, however, we offer a heuristic algorithm (one version for undirected graphs and one version for directed ones) to test if a nonplanar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently nondrawable.
Arc triangulations
 PROC. 26TH EUR. WORKSH. COMP. GEOMETRY (EUROCG’10)
, 2010
"... The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alter ..."
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Cited by 27 (2 self)
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The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.
Curvilinar graph drawing using the forcedirected method
 Proc. 12th Int. Symposium on Graph Drawing, 2004, Springer LNCS 3383
"... Abstract. We present a method for modifying a forcedirected graph drawing algorithm into an algorithm for drawing graphs with curved lines. Our method is based on embedding control points as dummy vertices so that edges can be drawn as splines. Our experiments show that our method yields aesthetica ..."
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Cited by 16 (0 self)
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Abstract. We present a method for modifying a forcedirected graph drawing algorithm into an algorithm for drawing graphs with curved lines. Our method is based on embedding control points as dummy vertices so that edges can be drawn as splines. Our experiments show that our method yields aesthetically pleasing curvilinear drawing with improved angular resolution. Applying our method to the GEM algorithm on the test suite of the “Rome Graphs ” resulted in an average improvement of 46 % in angular resolution and of almost 6 % in edge crossings. 1
Improving Angular Resolution in Visualizations of Geographic Networks
, 2000
"... In visualizations of largescale transportation and communications networks, node coordinates are usually fixed to preserve the underlying geography, while links are represented as geodesics for simplicity. This often leads ..."
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Cited by 13 (3 self)
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In visualizations of largescale transportation and communications networks, node coordinates are usually fixed to preserve the underlying geography, while links are represented as geodesics for simplicity. This often leads
Planar and PolyArc Lombardi Drawings
"... Abstract. 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 kLombardi drawings, in which ..."
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Cited by 4 (4 self)
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Abstract. 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 kLombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2Lombardi drawing. We show that every planar graph has a smooth planar 3Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings. 1
L.: Forcedirected Lombardistyle graph drawing
 In: Proc. 19th Int. Symp. on Graph Drawing
, 2011
"... Abstract. A Lombardi drawing of a graph is defined as one in which vertices are represented as points, edges are represented as circular arcs between their endpoints, and every vertex has perfect angular resolution (angles between consecutive edges, as measured by the tangents to the circular arcs a ..."
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Cited by 4 (2 self)
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Abstract. A Lombardi drawing of a graph is defined as one in which vertices are represented as points, edges are represented as circular arcs between their endpoints, and every vertex has perfect angular resolution (angles between consecutive edges, as measured by the tangents to the circular arcs at the vertex, all have the same degree). We describe two algorithms that create “Lombardistyle” drawings (which we also call nearLombardi drawings), in which all edges are still circular arcs, but some vertices may not have perfect angular resolution. Both of these algorithms take a forcedirected, springembedding approach, with one using forces at edge tangents to produce curved edges and the other using dummy vertices on edges for this purpose. As we show, these approaches both produce nearLombardi drawings, with one being slightly better at achieving nearperfect angular resolution and the other being slightly better at balancing vertex placements. 1
Fast Layout Methods for Timetable Graphs
 Proceedings of Graph Drawing 2000, Lecture Notes in Computer Science
, 2000
"... Timetable graphs are used to analyze transportation networks. In their visualization, vertex coordinates are xed to preserve the underlying geography, but due to small angles and overlaps, not all edges should be represented by geodesics (straight lines or great circles). A previously introduced ..."
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Cited by 3 (1 self)
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Timetable graphs are used to analyze transportation networks. In their visualization, vertex coordinates are xed to preserve the underlying geography, but due to small angles and overlaps, not all edges should be represented by geodesics (straight lines or great circles). A previously introduced algorithm represents a subset of the edges by Bezier curves, and places control points of these curves using a forcedirected approach [5]. While the results are of very good quality, the running times make the approach impractical for interactive systems. In this paper, we present a fast layout algorithm using an entirely different approach to edge routing, based on directions of control segments rather than positions of control points. We reveal an interesting theoretical connection with Tutte's barycentric layout method [18], and our computational studies show that this new approach yields satisfactory layouts even for huge timetable graphs within seconds. 1
Polar Coordinate Drawing of Planar Graphs with Good Angular Resolution
 In Proc. 9th Symposium on Graph Drawing
, 2003
"... We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms for constructing it. The main advantage of the polar representation is that it allows independent control over grid size and bend positions. ..."
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Cited by 3 (2 self)
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We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms for constructing it. The main advantage of the polar representation is that it allows independent control over grid size and bend positions. We first describe a standard (Cartesian) representation algorithm, CRA, which we then modify to obtain a polar representation algorithm, PRA. In both algorithms we are concerned with the following drawing criteria: angular resolution, bends per edge, vertex resolution, bendpoint resolution, edge separation, and drawing area. The CRA algorithm achieves...
Drawing graphs with large vertices and thick edges
 Proc. 8th Workshop on Algorithms and Data Structures
, 2003
"... Abstract. We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient lineartime algorithm which draws edges a ..."
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Cited by 2 (1 self)
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Abstract. We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient lineartime algorithm which draws edges and vertices of varying 2dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid. 1
Smooth Orthogonal Layouts
"... Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axisaligned line segments, in smooth orthogonal layouts every edge is made of axisaligned segments and circular arcs with common tange ..."
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Cited by 1 (1 self)
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Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axisaligned line segments, in smooth orthogonal layouts every edge is made of axisaligned segments and circular arcs with common tangents. Our goal is to create such layouts with low edge complexity, measured by the number of line and circular arc segments. We show that every biconnected 4planar graph has a smooth orthogonal layout with edge complexity 3. If the input graph has a complexity2 traditional orthogonal layout we can transform it into a smooth complexity2 layout. Using the Kandinsky model for removing the degree restriction, we show that any planar graph has a smooth complexity2 layout. 1