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19
Drawing Trees with Perfect Angular Resolution and Polynomial Area
"... Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossingfree straightline drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require e ..."
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Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossingfree straightline drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossingfree straightline drawing having perfect angular resolution. 3. Any ordered tree has a crossingfree Lombardistyle 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 straightline drawings and what more is achievable with Lombardistyle drawings, with respect to drawings of trees with perfect angular resolution. 1
Forcedirected Lombardistyle graph drawing
 IN: PROC. 19TH INT. SYMP. ON GRAPH DRAWING
, 2011
"... 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 ve ..."
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Cited by 10 (4 self)
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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.
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 7 (4 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
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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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
Spring embedders and force directed graph drawing algorithms,” arXiv preprint arXiv:1201.3011
, 2012
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On The Usability of Lombardi Graph Drawings
"... Abstract. A recent line of work in graph drawing studies Lombardi drawings, i.e., drawings with circulararc edges and perfect angular resolution at vertices. Little has been known about the effects of curved edges versus straight edges in typical graph reading tasks. In this paper we present the fi ..."
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Cited by 3 (0 self)
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Abstract. A recent line of work in graph drawing studies Lombardi drawings, i.e., drawings with circulararc edges and perfect angular resolution at vertices. Little has been known about the effects of curved edges versus straight edges in typical graph reading tasks. In this paper we present the first user evaluation that empirically measures the readability of three different layout algorithms (traditional spring embedder and two recent nearLombardi forcebased algorithms) for three different tasks (shortest path, common neighbor, vertex degree). The results indicate that, while users prefer the Lombardi drawings, the performance data do not present such a positive picture. 1
Optical Graph Recognition
"... Optical graph recognition (OGR) reverses graph drawing. A drawing transforms the topological structure of a graph into a graphical representation. Primarily, it maps vertices to points and displays them by icons and it maps edges to Jordan curves connecting the endpoints. OGR transforms the digital ..."
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Cited by 2 (2 self)
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Optical graph recognition (OGR) reverses graph drawing. A drawing transforms the topological structure of a graph into a graphical representation. Primarily, it maps vertices to points and displays them by icons and it maps edges to Jordan curves connecting the endpoints. OGR transforms the digital image of a drawn graph into its topological structure. It consists of four phases, preprocessing, segmentation, topology recognition, and postprocessing. OGR is based on established digital image processing techniques. Its novelty is the topology recognition where the edges are recognized with emphasis on the attachment to their vertices and on edge crossings. Our prototypical implementation OGR up shows the effectiveness of the approach and produces a GraphML file which can be used for further algorithmic studies and graph drawing tools.
The Graphs of Planar Soap Bubbles
, 2012
"... We characterize the graphs formed by twodimensional soap bubbles as being exactly the 3regular bridgeless planar multigraphs. Our characterization combines a local characterization of soap bubble graphs in terms of the curvatures of arcs meeting at common vertices, a proof that this characterizati ..."
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Cited by 2 (1 self)
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We characterize the graphs formed by twodimensional soap bubbles as being exactly the 3regular bridgeless planar multigraphs. Our characterization combines a local characterization of soap bubble graphs in terms of the curvatures of arcs meeting at common vertices, a proof that this characterization remains invariant under Möbius transformations, an application of Möbius invariance to prove bridgelessness, and a Möbiusinvariant power diagram of circles previously developed by the author for its applications in graph drawing.
Triangulations with Circular Arcs
"... An important objective in the choice of a triangulation is that the smallest angle becomes as large as possible. In the straightline case, it is known that the Delaunay triangulation is optimal in this respect. We propose and study the concept of a circular arc triangulation— a simple and effective ..."
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An important objective in the choice of a triangulation is that the smallest angle becomes as large as possible. In the straightline case, it is known that the Delaunay triangulation is optimal in this respect. We propose and study the concept of a circular arc triangulation— a simple and effective alternative that offers flexibility for additionally enlarging small angles—and discuss its applications in graph drawing.
Social Networks
, 2005
"... Social networks provide a rich source of graph drawing problems, because they appear in an incredibly wide variety of forms and contexts. After sketching the scope of social network analysis, we establish some general principles for social network visualization before finally reviewing applications ..."
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Social networks provide a rich source of graph drawing problems, because they appear in an incredibly wide variety of forms and contexts. After sketching the scope of social network analysis, we establish some general principles for social network visualization before finally reviewing applications of, and challenges for, graph drawing methods in this area. Other accounts more generally relating to social network visualization are given, e.g., in [Klo81, BKR + 99a, Fre00, Fre05, BKR06].