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18
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 13 (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.
Crossing and weighted crossing number of nearplanar graphs, Algorithmica
"... A nonplanar graph G is nearplanar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a nearplanar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop minmax formulas involving efficiently computable lower and u ..."
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Cited by 11 (2 self)
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A nonplanar graph G is nearplanar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a nearplanar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop minmax formulas involving efficiently computable lower and upper bounds. These minmax results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněn´y and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NPhard to compute a weighted version of the crossing number for nearplanar graphs. 1
Integrating edge routing into forcedirected layout
 IN: PROC. 14TH INTL. SYMP. GRAPH DRAWING (GD ’06). VOLUME 4372 OF LECTURE
, 2007
"... Forcedirected layout is typically used to create organiclooking, straightedge drawings of large graphs while combinatorial techniques are generally preferred for highquality layout of small to medium sized graphs. In this paper we integrate edgerouting techniques into a forcedirected layout me ..."
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Cited by 7 (4 self)
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Forcedirected layout is typically used to create organiclooking, straightedge drawings of large graphs while combinatorial techniques are generally preferred for highquality layout of small to medium sized graphs. In this paper we integrate edgerouting techniques into a forcedirected layout method based on constrained stress majorisation. Our basic procedure takes an initial layout for the graph, including polyline paths for the edges, and improves this layout by moving the nodes to reduce stress and moving edge bend points to straighten the edges and reduce their overall length. Separation constraints between nodes and edge bend points are used to ensure that node labels do not overlap edges or other nodes and that no additional edge crossings are introduced.
Topology Preserving Constrained Graph Layout
"... Abstract. Constrained graph layout is a recent generalisation of forcedirected graph layout which allows constraints on node placement. We give a constrained graph layout algorithm that takes an initial feasible layout and improves it while preserving the topology of the initial layout. The algorith ..."
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Cited by 7 (3 self)
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Abstract. Constrained graph layout is a recent generalisation of forcedirected graph layout which allows constraints on node placement. We give a constrained graph layout algorithm that takes an initial feasible layout and improves it while preserving the topology of the initial layout. The algorithm supports polyline connectors and clusters. During layout the connectors and cluster boundaries act like impervious rubberbands which try to shrink in length. The intended application for our algorithm is dynamic graph layout, but it can also be used to improve layouts generated by other graph layout techniques. 1
An Experimental Comparison of Orthogonal Compaction Algorithms
 In Graph Drawing (Proc. GD 2000
, 2000
"... We present an experimental study in which we compare the stateoftheart methods for compacting orthogonal graph layouts. Given the shape of a planar orthogonal drawing, the task is to place the vertices and the bends on grid points so that the total area or the total edge length is minimised. We c ..."
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Cited by 7 (2 self)
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We present an experimental study in which we compare the stateoftheart methods for compacting orthogonal graph layouts. Given the shape of a planar orthogonal drawing, the task is to place the vertices and the bends on grid points so that the total area or the total edge length is minimised. We compare four constructive heuristics based on rectangular dissection and on turnregularity, also in combination with two improvement heuristics based on longest paths and network flows, and an exact method which is able to compute provable optimal drawings of minimum total edge length. We provide a performance evaluation in terms of quality and running time. The test data consists of two testsuites already used in previous experimental research. In order to get hard instances, we randomly generated an additional set of planar graphs.
Crossing minimization meets simultaneous drawing
 In IEEE Pacific Visualisation Symposium
, 2008
"... We define the concept of crossing numbers for simultaneous graphs by extending the crossing number problem of traditional graphs. We discuss differences to the traditional crossing number problem, and give an NPcompleteness proof and lower and upper bounds for the new problem. Furthermore, we show ..."
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Cited by 7 (3 self)
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We define the concept of crossing numbers for simultaneous graphs by extending the crossing number problem of traditional graphs. We discuss differences to the traditional crossing number problem, and give an NPcompleteness proof and lower and upper bounds for the new problem. Furthermore, we show how existing heuristic and exact algorithms for the traditional problem can be adapted to the new task of simultaneous crossing minimization, and report on a brief experimental study of their implementations.
Planarity testing and optimal edge insertion with embedding constraints
, 2008
"... The planarization method has proven to be successful in graph drawing. The output, a combinatorial planar embedding of the socalled planarized graph, can be combined with stateoftheart planar drawing algorithms. However, many practical applications have additional constraints on the drawings tha ..."
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Cited by 6 (2 self)
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The planarization method has proven to be successful in graph drawing. The output, a combinatorial planar embedding of the socalled planarized graph, can be combined with stateoftheart planar drawing algorithms. However, many practical applications have additional constraints on the drawings that result in restrictions on the set of admissible planar embeddings. In this paper, we consider embedding constraints that restrict the admissible order of incident edges around a vertex. Such constraints occur in applications, e.g., from side or port constraints. We introduce a set of hierarchical embedding constraints that include grouping, oriented, and mirror constraints, and show how these constraints can be integrated into the planarization method. For this, we first present a linear time algorithm for testing if a given graph G is ecplanar, i.e., admits a planar embedding satisfying the given embedding constraints. In the case that G is ecplanar, we provide a linear time algorithm for computing the corresponding ecembedding. Otherwise, an ecplanar subgraph is computed. The critical part is to reinsert the deleted edges subject to the embedding constraints so that the number of crossings is kept small. For this, we present a linear time algorithm which is able to insert an edge into an ecplanar graph H so that the insertion is crossing minimal among all ecplanar embeddings of H. As a side result, we characterize the set of all possible ecplanar embeddings using BC and SPQRtrees.
Algorithms for the hypergraph and the minor crossing number problems
 In Proc. ISAAC’07, volume 4835 of LNCS
, 2007
"... Abstract. We consider the problems of hypergraph and minor crossing minimization, and point out a relation between these two problems which has not been exploited before. We present some complexity results regarding the corresponding edge and node insertion problems. Based on these results, we give ..."
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Cited by 6 (4 self)
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Abstract. We consider the problems of hypergraph and minor crossing minimization, and point out a relation between these two problems which has not been exploited before. We present some complexity results regarding the corresponding edge and node insertion problems. Based on these results, we give the first embeddingbased heuristics to tackle both problems and present a short experimental study. Furthermore, we give the first exact ILP formulation for both problems. 1
Inserting a vertex into a planar graph
 In ACMSIAM Symposium on Discrete Algorithms 2009; ACM Press
, 2009
"... We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u)  u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a plana ..."
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Cited by 6 (6 self)
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We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u)  u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a planar embedding of G, in which the given star can be inserted requiring the minimum number of crossings. This is a generalization of the crossing minimum edge insertion problem [15], and can help to find improved approximations for the crossing minimization problem. Indeed, in practice, the algorithm for the crossing minimum edge insertion problem turned out to be the key for obtaining the currently strongest approximate solutions for the crossing number of general graphs. The generalization considered here can lead to even better solutions for the crossing minimization problem. Furthermore, it offers new insight into the crossing number problem for almostplanar and apex graphs. It has been an open problem whether the star insertion problem is polynomially solvable. We give an affirmative answer by describing the first efficient algorithm for this problem. This algorithm uses the SPQRtree data structure to handle the exponential number of possible embeddings, in conjunction with dynamic programming schemes for which we introduce partitioning cost subproblems. 1
NonPlanar Core Reduction of Graphs
"... Abstract. We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be compu ..."
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Cited by 5 (5 self)
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Abstract. We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs. 1