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On the Computational Complexity of Upward and Rectilinear Planarity Testing (Extended Abstract)
, 1994
"... A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical se ..."
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Cited by 84 (4 self)
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A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment, and no two edges cross. Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures. In this paper we show that upward planarity testing and rectilinear planarity testing are NPcomplete problems. We also show that it is NPhard to approximate the minimum number of bends in a planar orthogonal drawing of an nvertex graph with an O(n 1\Gammaffl ) error, for any ffl ? 0.
Upward Planarity Testing
 SIAM Journal on Computing
, 1995
"... Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upw ..."
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Cited by 78 (14 self)
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Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and singlesource digraphs. We also sketch the proof of NPcompleteness of upward planarity testing.
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 25 (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...
A New Approach for Visualizing UML Class Diagrams
"... UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combina ..."
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Cited by 14 (0 self)
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UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combination of hierarchical and nonhierarchical relations poses a special challenge to a graph layout tool. Existing layout tools treat hierarchical and nonhierarchical relations either alike or as separate tasks in a twophase process as in, e.g., [Seemann 1997]. We suggest a new approach for visualizing UML class diagrams leading to a balanced mixture of the following aesthetic criteria: Crossing minimization, bend minimization, uniform direction within each class hierarchy, no nesting of one class hierarchy within another, orthogonal layout, merging of multiple inheritance edges, and good edge labelling. We have realized our approach within the graph drawing library GoVisual. Experiments show the superiority to stateoftheart and industrial standard layouts.
Graph Drawing
 Lecture Notes in Computer Science
, 1997
"... INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducte ..."
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Cited by 14 (3 self)
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INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and humancomputer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin
Comparing trees via crossing minimization
 In Proc. 25th Conf. on Foundations of Software Technology and Theoretical Computer Science, volume 3821 of LNCS
, 2005
"... Abstract. Two trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a onetoone correspondence between pairs of leaves of the different trees. Do there exist two ..."
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Cited by 11 (1 self)
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Abstract. Two trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a onetoone correspondence between pairs of leaves of the different trees. Do there exist two planar embeddings of the two trees that minimize the crossings of the matching edges? This problem has important applications in the construction and evaluation of phylogenetic trees.
Layerfree upward crossing minimization
 ACM Journal of Experimental Algorithmics
"... Abstract. An upward drawing of a DAG G is a drawing of G in which all edges are drawn as curves increasing monotonically in the vertical direction. In this paper, we present a new approach for upward crossing minimization, i.e., finding an upward drawing of a DAG G with as few crossings as possible. ..."
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Cited by 10 (7 self)
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Abstract. An upward drawing of a DAG G is a drawing of G in which all edges are drawn as curves increasing monotonically in the vertical direction. In this paper, we present a new approach for upward crossing minimization, i.e., finding an upward drawing of a DAG G with as few crossings as possible. Our algorithm is based on a twostage upward planarization approach, which computes a feasible upward planar subgraph in the first step, and reinserts the remaining edges by computing constraintfeasible upward insertion paths. An experimental study shows that the new algorithm leads to much better results than existing algorithms for upward crossing minimization, including the classical Sugiyama approach. 1
Algorithm and Experiments in Testing Planar Graphs for Isomorphism
, 2004
"... We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determi ..."
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Cited by 6 (0 self)
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We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determine the conditions in which the implemented algorithm outperforms other graph matchers, which do not impose topological restrictions on graphs. We report experiments with our planar graph matcher tested against McKay’s, Ullmann’s, and SUBDUE’s (a graphbased data mining system) graph matchers.
Drawing Binary Tanglegrams: An Experimental Evaluation
, 2009
"... A tanglegram is a pair of trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. In applications such as phylogenetics or hierarchical clustering, it is required that the individual trees are drawn crossingfree. A natural optimization problem, deno ..."
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Cited by 6 (2 self)
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A tanglegram is a pair of trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. In applications such as phylogenetics or hierarchical clustering, it is required that the individual trees are drawn crossingfree. A natural optimization problem, denoted tanglegram layout problem, is thus to minimize the number of crossings between intertree edges. The tanglegram layout problem is NPhard even for complete binary trees, for general binary trees the problem is hard to approximate if the Unique Games Conjecture holds. In this paper we present an extensive experimental comparison of a new and several known heuristics for the general binary case. We measure the performance of the heuristics with a simple integer linear program and a new exact branchandbound algorithm. The new heuristic returns the first solution that the branchandbound algorithm computes (in quadratic time). Surprisingly, in most cases this simple heuristic is at least as good as the best of the other heuristics.
A.: Drawing (complete) binary tanglegrams: Hardness, approximation, fixedparameter tractability. Arxiv report
, 2008
"... Abstract. A binary tanglegram is a pair 〈S, T 〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossings and that the intertre ..."
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Cited by 6 (2 self)
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Abstract. A binary tanglegram is a pair 〈S, T 〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossings and that the intertree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NPhard and that the problem is fixedparameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constantfactor approximation for general binary trees. We show that the problem is hard even if both trees are complete binary trees. For this case we give an O(n 3)time 2approximation and a new and simple fixedparameter algorithm. We show that the maximization version of the dual problem for general binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878approximation. 1