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Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
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Cited by 78 (0 self)
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We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
StraightLine Drawing Algorithms for Hierarchical Graphs and Clustered Graphs
 Algorithmica
, 1999
"... Hierarchical graphs and clustered graphs are useful nonclassical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualizatio ..."
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Cited by 71 (12 self)
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Hierarchical graphs and clustered graphs are useful nonclassical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straightline representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straightline hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straightline drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.
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 53 (9 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.
Effective information visualisation: a study of graph Drawing Aesthetics . . .
, 2000
"... Information visualisation systems which generate diagrams representing discrete relational information must consider potential users if they are to be effective. Many algorithms which render an abstract graph structure as a diagram are valued for their conformance to aesthetic criteria (e.g. reducin ..."
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Cited by 29 (2 self)
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Information visualisation systems which generate diagrams representing discrete relational information must consider potential users if they are to be effective. Many algorithms which render an abstract graph structure as a diagram are valued for their conformance to aesthetic criteria (e.g. reducing the number of edge crossings, maximising symmetry), or for computational efficiency. They are not usually judged on their ability to produce diagrams that maximise human performance. This paper presents the results of experiments investigating the relative worth (from an HCI point of view) of graph drawing aesthetics and algorithms using a single graph. The results indicate that while some individual aesthetics affect human performance, it is difficult to say that one algorithm is `better' than another from a relational understanding point of view. Designers of automatic layout algorithms, and the systems which embody such algorithms, can benefit from this study and this humancentred approach, by adapting their methods to focus on user concerns, rather than computa
Algorithms for Drawing Clustered Graphs
, 1997
"... In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics ..."
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Cited by 29 (2 self)
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In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics workstation as graph drawings. The usefulness of the relational model depends on whether the graph drawings effectively convey the relational information to the users. This thesis is concerned with finding good drawings of graphs. As the amount of information that we want to visualize becomes larger and the relations become more complex, the classical graph model tends to be inadequate. Many extended models use a node hierarchy to help cope with the complexity. This thesis introduces a new graph model called the clustered graph. The central theme of the thesis is an investigation of efficient algorithms to produce good drawings for clustered graphs. Although the criteria for judging the qua...
Comparing and Evaluating Layout Algorithms within GraphEd
 J. Visual Languages and Computing
, 1995
"... This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd ..."
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Cited by 19 (2 self)
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This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd
Drawing High Degree Graphs with Low Bend Numbers
 PROC. 4TH SYMPOSIUM ON GRAPH DRAWING (GD'95), LNCS 1027
, 1995
"... We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any respect to the size of the vertices. Naturally, the vertice ..."
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Cited by 15 (2 self)
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We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any respect to the size of the vertices. Naturally, the vertices should be represented by uniformly small squares. In addition we might require that each face should be represented by a nonempty region. This would allow a labeling of the faces. We present an efficient algorithm which provably achieves the bend minimum following these constraints. Omitting the latter requirement we conjecture that the problem becomes NPhard. For that case, we give advices for good approximations. We demonstrate the effectiveness of our approaches giving some interesting examples.
An Algorithm for StraightLine Drawing of Planar Graphs
, 1995
"... Abstract. We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a ..."
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Abstract. We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs “nicely, ” as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n − 4) × (n − 2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Key Words.
Randomized Graph Drawing with HeavyDuty Preprocessing
 In: AVI ’94: Proceedings of the Workshop on Advanced Visual Interfaces
, 1994
"... : We present a graph drawing system for general undirected graphs with straightline edges. It carries out a rather complex set of preprocessing steps, designed to produce a topologically good, but not necessarily nicelooking layout, which is then subjected to Davidson and Harel's simulated an ..."
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Cited by 14 (1 self)
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: We present a graph drawing system for general undirected graphs with straightline edges. It carries out a rather complex set of preprocessing steps, designed to produce a topologically good, but not necessarily nicelooking layout, which is then subjected to Davidson and Harel's simulated annealing beautification algorithm. The intermediate layout is planar for planar graphs and attempts to come close to planar for nonplanar graphs. The system's results are significantly better, and much faster, than what the annealing approach is able to achieve on its own. 1 Introduction A large amount of work on the problem of graph layout has been carried out in recent years, resulting in a number of sophisticated and powerful algorithms. An extensive and detailed survey can be found in [BETT93]. Many of the approaches taken are limited to special cases of graphs, such as trees or planar graphs; others concentrate on special kinds of layouts, such as rectilinear grid drawings, or convex drawin...