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23
Orthogonal Drawings Based On The Stratification Of Planar Graphs
, 2000
"... Several algorithms have been proposed to draw planar graphs using 2visibility and Kandinsky Models. Here, we propose three new algorithms implementing these models in linear time using small grid sizes and few bends. These algorithms are all based on the construction of a particular layered spannin ..."
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Cited by 4 (2 self)
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Several algorithms have been proposed to draw planar graphs using 2visibility and Kandinsky Models. Here, we propose three new algorithms implementing these models in linear time using small grid sizes and few bends. These algorithms are all based on the construction of a particular layered spanning tree called Stratification. A linear time algorithm that computes a stratification is also presented.
New Lower Bounds For Orthogonal Drawings
 J. Graph Algorithms Appl
, 1998
"... An orthogonal drawing of a graph is an embedding of the graph in the twodimensional grid such that edges are routed along gridlines. In this paper we explore lower bounds for orthogonal graph drawings. We prove lower bounds on the number of bends and, when crossings are not allowed, also lower bou ..."
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Cited by 3 (0 self)
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An orthogonal drawing of a graph is an embedding of the graph in the twodimensional grid such that edges are routed along gridlines. In this paper we explore lower bounds for orthogonal graph drawings. We prove lower bounds on the number of bends and, when crossings are not allowed, also lower bounds on the size of the grid. Communicated by D. Wagner: submitted July 1997; revised November 1998. Some results of this paper were part of the author's diploma thesis at TU Berlin under the supervision of Prof. R. Mohring, and have been presented in an extended abstract at Graph Drawing '95, Passau, Germany. T. Biedl., New Lower Bounds, JGAA, 2(7) 131 (1998) 2 1 Introduction A graph G = (V; E) is an abstract structure consisting of points (or vertices) V and connections (or edges) E. Such a structure is found in many industrial applications, such as networks, production schedules and diagrams. With the aid of graph drawing, a graph is displayed in visual form, and the underlying infor...
Orthogonal drawings with few layers
 PROC. 9TH INTERNATIONAL SYMP. ON GRAPH DRAWING (GD '01
, 2002
"... In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very smal ..."
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Cited by 3 (2 self)
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In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain pointdrawings with 3layers and 4 bends per edge, and algorithms to obtain boxdrawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds.
Visualizing Software for Telecommunication Services
"... An active research area in telecommunications concerns how to specify and control the addition of new services, such as call waiting or instant messaging, into existing software. One approach is to rely on a componentbased architecture such as Distributed Feature Composition (DFC), by which a new s ..."
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Cited by 2 (0 self)
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An active research area in telecommunications concerns how to specify and control the addition of new services, such as call waiting or instant messaging, into existing software. One approach is to rely on a componentbased architecture such as Distributed Feature Composition (DFC), by which a new service can be specified as a composition of primitive features over time. Formally, a communication episode is represented by a dynamic graph of software feature boxes, called a usage. This serves as the fundamental model for how services are invoked and how they interact with other services. This paper, after providing some background on DFC, discusses a technique for visualizing the usages which arise through DFC specifications. With the visualization, users can monitor and validate service protocols and feature interactions in real time or through playback logs. The principal display component uses a novel variation of forcedirected layouts for undirected graphs. The resulting graphical interface has become a principal tool for developers building services using DFC.
Efficient Algorithms for Drawing Planar Graphs
, 1999
"... x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ................. ..."
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x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ............................ 6 1.2.4 Orthogonal drawings . . ........................... 7 1.2.5 Grid drawings ................................ 8 1.3 Properties of Drawings ................................ 9 1.4 Scope of this Thesis .................................. 10 1.4.1 Rectangular drawings . . . ......................... 11 1.4.2 Orthogonal drawings . . ........................... 12 1.4.3 Boxrectangular drawings ........................... 14 1.4.4 Convex drawings . . ............................. 16 1.5 Summary ....................................... 16 2 Preliminaries 20 2.1 Basic Terminology .................................. 20 2.1.1 Graphs and Multigraphs ........................... 20 i CO...
Minimising the Number of Bends and Volume in 3Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout
, 2004
"... A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3d ..."
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A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, socalled diagonal drawings. This vertexlayout strategy was introduced in the 3BENDS algorithm of Eades et al. [Discrete Applied Math. 103:5587, 2000]. We show that minimising the number of bends in a diagonal drawing of a given graph is NPhard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal in linear time. Using two heuristics for determining this vertexordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the abovementioned 3BENDS algorithm, produces 3bend drawings with o(n ) volume, which is the best known upper bound for the volume of 3dimensional orthogonal graph drawings with at most three bends per edge.
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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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
Interactive Graph Drawing Algorithms
"... Traditional graph drawing algorithms can also be called oneshot algorithms because given a graph as input, they typically construct a drawing only once. However, in several applications, such as software engineering and database design, users interact extensively with a displayed graph, continuousl ..."
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Traditional graph drawing algorithms can also be called oneshot algorithms because given a graph as input, they typically construct a drawing only once. However, in several applications, such as software engineering and database design, users interact extensively with a displayed graph, continuously adding or deleting its vertices and edges. In such a scenario, a graph drawing system should update a drawing each time the displayed graph is updated by the user. Unfortunately, traditional drawing algorithms may not be suitable in such situations. Since they typically construct a drawing from scratch, they may fail to update a drawing quickly after the user updates the displayed graph. Also, the new drawing constructed after the update may be significantly different from the previous drawing even if only a small change was made in the displayed graph, making it difficult for the user to correlate the previous drawing and the new drawing (making only a small change in a drawing when a sma...
Fully Dynamic Orthogonal Graph Layout for Interactive Systems
, 2000
"... . We combine and supplement existing approaches to arrive at a general algorithm for dynamic orthogonal graph drawing with few bends. Our approach is applicable to nonplanar graphs with vertices of arbitrary degree, reduces the number of bends if possible, allows explicit control of the bendnu ..."
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. We combine and supplement existing approaches to arrive at a general algorithm for dynamic orthogonal graph drawing with few bends. Our approach is applicable to nonplanar graphs with vertices of arbitrary degree, reduces the number of bends if possible, allows explicit control of the bendnumber/modication tradeo and does not suer from the socalled rotation problem, all of which have been limitations in earlier approaches. 1 Introduction Orthogonal representations are traditionally used for graphs in technical applications like network plans, EntityRelationship diagrams, program dependencies, or circuit schematics. Support software like CASE tools or schema editors typically allows interactive editing of these graphs, but burdens the user with the task to provide a readable layout. Instead of relying on manual placement, an interactive system would ideally oer the kind of transparent layout facilities known from word processing systems that dynamically adjust the for...
BoundedDegree Book Embedings and ThreeDimensional Orthogonal Graph Drawing
 Proc. 9th International Symp. on Graph Drawing (GD '01), volume 2265 of Lecture Notes in Comput. Sci
, 2002
"... A book embedding of a graph consists of a linear orderin of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph $G=(V,E)$ ..."
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A book embedding of a graph consists of a linear orderin of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph $G=(V,E)$, let $f:V\rightarrow\mathbb{N}$ be a function such that $1\leq f(v)\leq\deg(v)$. We present a Las Vegas algorithm which produces a book embedding of $G$ with \Oh{\sqrt{E\cdot\max_v\ceil{\deg(v)/f(v)}}} pages, such that at most $f(v)$ edges incident to a vertex $v$ are on a single page. This algorithm generalises existing results for book embeddings. We apply this algorithm to produce 3D orthogonal drawings with one bend per edge and \Oh{V^{3/2}E} volume, and \emph{singlerow} drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some $Z$plane; such drawings are without socalled \emph{crosscuts}, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with \Oh{VE} volume but with crosscuts. These results establish improved bounds for the volume of 3D orthogonal graph drawings.