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17
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 59 (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.
A Fast Effective Heuristic For The Feedback Arc Set Problem
 Information Processing Letters
, 1993
"... Let G = (V; A) denote a simple connected directed graph, and let n = jV j, m = jAj, where n \Gamma 1 m \Gamma n 2 \Delta . A feedback arc set (FAS) of G, denoted R(G), is a (possibly empty) set of arcs whose reversal makes G acyclic. A minimum feedback arc set of G, denoted R (G), is a FAS ..."
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Cited by 24 (0 self)
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Let G = (V; A) denote a simple connected directed graph, and let n = jV j, m = jAj, where n \Gamma 1 m \Gamma n 2 \Delta . A feedback arc set (FAS) of G, denoted R(G), is a (possibly empty) set of arcs whose reversal makes G acyclic. A minimum feedback arc set of G, denoted R (G), is a FAS of minimum cardinality r (G); the computation of R (G) is called the FAS problem. For every n, let ae(n) denote the maximum, over all digraphs G of order n, of jR (G)j. Berger and Shor have recently published an algorithm which, for a given digraph G, computes a FAS whose cardinality is O(ae(n)). Thus the Berger/Shor algorithm provides, in a certain asymptotic sense, an optimal solution to the FAS problem. Unfortunately, the Berger/Shor algorithm is complicated and requires running time O(mn). In this paper we present a simple FAS algorithm which guarantees a good (though not optimal) performance bound and executes in time O(m). Further, for the sparse graphs which arise frequentl...
Spring Algorithms and Symmetry
 Theoretical Computer Science
, 1999
"... Spring algorithms are regarded as effective tools for visualizing undirected graphs. One major feature of applying spring algorithms is to display symmetric properties of graphs. This feature has been confirmed by numerous experiments. In this paper, firstly we formalize the concepts of graph symmet ..."
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Cited by 22 (3 self)
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Spring algorithms are regarded as effective tools for visualizing undirected graphs. One major feature of applying spring algorithms is to display symmetric properties of graphs. This feature has been confirmed by numerous experiments. In this paper, firstly we formalize the concepts of graph symmetries in terms of "reflectional" and "rotational" automorphisms; and characterize the types of symmetries, which can be displayed simultaneously by a graph layout, in terms of "geometric" automorphism groups. We show that our formalization is complete. Secondly, we provide general theoretical evidence of why many spring algorithms can display graph symmetry. Finally, the strength of our general theorem is demonstrated from its application to several existing spring algorithms. 1 Introduction Graphs are commonly used in Computer Science to model relational structures such as programs, databases, and data structures. A good graph "layout" gives a clear understanding of a structural model; a ba...
Using Genetic Algorithms for Drawing Undirected Graphs
 The Third Nordic Workshop on Genetic Algorithms and their Applications
, 1996
"... In this paper we report on our experiences with applying genetic algorithms to the drawing of undirected graphs with straight line edges. Since there exists a relatively simple but powerful heuristic for this class of graphs, namely the spring algorithm, we use this algorithm as a local finetuner w ..."
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Cited by 20 (1 self)
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In this paper we report on our experiences with applying genetic algorithms to the drawing of undirected graphs with straight line edges. Since there exists a relatively simple but powerful heuristic for this class of graphs, namely the spring algorithm, we use this algorithm as a local finetuner within the genetic algorithm. We compare our results with drawings produced by the spring algorithm alone and discuss the strengths and weaknesses of the approach presented. Keywords: spring algorithm, evolutionary algorithm, genetic algorithm, graph drawing 1 Introduction The problem of drawing a graph nicely can be regarded as searching for an optimal layout of a given graph according to some measurable aesthetics. However, solving this problem to optimality seems to be computationally infeasible even for relatively simple aesthetic criteria [7], thus one is bound to the area of heuristics and stochastic search methods. In recent years, a lot of research on the problem to support the auto...
An Algorithm For Drawing A Hierarchical Graph
, 1995
"... this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1. ..."
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Cited by 18 (7 self)
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this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1.
Drawing planar graphs symmetrically, III: Oneconnected planar graphs
 ALGORITHMICA
, 2006
"... Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can b ..."
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Cited by 8 (5 self)
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Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NPcomplete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.
Graph multidrawing: Finding nice drawings without defining nice
 the Proc. of Graph Drawing
, 1998
"... Abstract. This paper proposes a multidrawing approach to graph drawing. Current graphdrawing systems typically produce only one drawing of a graph. By contrast, the multidrawing approach calls for systematically producing many drawings of the same graph, where the drawings presented to the user rep ..."
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Cited by 7 (1 self)
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Abstract. This paper proposes a multidrawing approach to graph drawing. Current graphdrawing systems typically produce only one drawing of a graph. By contrast, the multidrawing approach calls for systematically producing many drawings of the same graph, where the drawings presented to the user represent a balance between aesthetics and diversity. This addresses a fundamental problem in graph drawing, namely, how to avoid requiring the user to specify formally and precisely all the characteristics of a single “nice ” drawing. We present a proofofconcept implementation with which we produce diverse selections of symmetriclooking drawings for small graphs. 1
On the Computational Complexity of Edge Concentration
 DAMATH: Discrete Applied Mathematics and Combinatorial Operational Research
, 1999
"... Suppose that G = (U; L; E) is a bipartite graph with vertex set U [L and edge set E U L. A typical convention for drawing G is to put the vertices of U on a horizontal line and the vertices of L on another horizontal line, and then to represent edges by line segments between the vertices that d ..."
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Cited by 5 (1 self)
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Suppose that G = (U; L; E) is a bipartite graph with vertex set U [L and edge set E U L. A typical convention for drawing G is to put the vertices of U on a horizontal line and the vertices of L on another horizontal line, and then to represent edges by line segments between the vertices that determine them. \Edge concentration" is known as an eective method to draw dense bipartite graphs clearly. The key in the edge concentration method is to reduce the number of edges, while the graph structural information is retained. The problem of having a maximal reduction on the number of edges by the edge concentration method was left open. In this paper we show that this problem is NPhard. Keywords: Graph drawing, Bipartite graph, Edge cover, NPcomplete. 1 Introduction Graphs are commonly used in computer science to model relation structures such as programs, databases, and data structures. A good graph drawing gives a clear understanding of a structural model; a bad drawing...
Area Requirement for Drawing Hierarchically Planar Graphs
, 1997
"... Abstract. In this paper, we investigate area requirements for drawing st hierarchically planar graphs by straightlines. Two drawing standards will be discussed: 1) each vertex is represented by a point and 2) grid visibifity representation (that is, a line segment is allowed to represent a vertex) ..."
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Cited by 3 (1 self)
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Abstract. In this paper, we investigate area requirements for drawing st hierarchically planar graphs by straightlines. Two drawing standards will be discussed: 1) each vertex is represented by a point and 2) grid visibifity representation (that is, a line segment is allowed to represent a vertex). For the first drawing standard, we show an exponential area lower bound needed for drawing hierarchically planar graphs. The lower bound holds even for hierarchical graphs without transitive arcs, in contrast to the results for upward planar drawing. Applications of some existing algorithms from upward drawing can guarantee the quadratic drawing area for grid visibility representation but do not necessarily guarantee the minimum drawing area. Motivated by this, we will present another grid visibifity drawing algoiithm which is efficient and guarantees the minimum drawing area.
Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs
 Proceedings of the 5th International Conference on Computing and Combinatorics (COCOON ’99
, 1999
"... . Hierarchical graphs are an important class of graphs for modelling many real applications in software and information visualization. In this paper, we shall investigate the computational complexity of constructing minimum area grid visibility representations of hierarchically planar graphs. Fi ..."
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Cited by 1 (0 self)
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. Hierarchical graphs are an important class of graphs for modelling many real applications in software and information visualization. In this paper, we shall investigate the computational complexity of constructing minimum area grid visibility representations of hierarchically planar graphs. Firstly, we provide a quadratic algorithm that minimizes the drawing area with respect to a fixed planar embedding. This implies that the area minimization problem is polynomial time solvable restricted to the class of graphs whose planar embeddings are unique. Secondly, we show that the area minimization problem is generally NPhard. Keywords: Graph Drawing, Hierarchically Planar Graph, Visibility Representation, Drawing Area. 1 Introduction Automatic graph drawing plays an important role in many modern computerbased applications, such as CASE tools, software and information visualization, VLSI design, visual data mining, and internet navigation. Directed acyclic graphs are an importa...