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15
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 12 (8 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
An interactive multiuser system for simultaneous graph drawing
 In Proc. Intl. Symp. Graph Drawing
, 2004
"... Abstract. In this paper we consider the problem of simultaneous drawing of two graphs. The goal is to produce aesthetically pleasing drawings for the two graphs by means of a heuristic algorithm and with human assistance. Our implementation uses the DiamondTouch table, a multiuser, touchsensitive i ..."
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Cited by 9 (3 self)
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Abstract. In this paper we consider the problem of simultaneous drawing of two graphs. The goal is to produce aesthetically pleasing drawings for the two graphs by means of a heuristic algorithm and with human assistance. Our implementation uses the DiamondTouch table, a multiuser, touchsensitive input device, to take advantage of direct physical interaction of several users working collaboratively. The system can be downloaded at
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 9 (2 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.
Upward Planarization Layout
, 2011
"... Recently, we presented a new practical method for upward crossing minimization [8], which clearly outperformed existing approaches for drawing hierarchical graphs in that respect. The outcome of this method is an upward planar representation (UPR), a planarly embedded graph in which crossings are re ..."
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Cited by 7 (4 self)
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Recently, we presented a new practical method for upward crossing minimization [8], which clearly outperformed existing approaches for drawing hierarchical graphs in that respect. The outcome of this method is an upward planar representation (UPR), a planarly embedded graph in which crossings are represented by dummy vertices. However, straightforward approaches for drawing such UPRs lead to quite unsatisfactory results. In this paper, we present a new algorithm for drawing UPRs that greatly improves the layout quality, leading to good hierarchal drawings with few crossings. We analyze its performance on wellknown benchmark graphs and compare it with alternative approaches.
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 7 (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.
Approximating the Crossing Number of Graphs Embeddable In Any Orientable Surface
, 2010
"... The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrar ..."
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Cited by 6 (4 self)
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The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrary fixed orientable surface. Our approach combines some known tools with a powerful new lower bound on the crossing number of an embedded graph. This result extends previous results that gave such approximations in particular cases of projective, toroidal or apex graphs; it is a qualitative improvement over previously published algorithms that constructed lowcrossingnumber drawings of embeddable graphs without giving any approximation guarantees. No constant factor approximation algorithms for the crossing number problem over comparably rich classes of graphs are known to date.
Experiments on exact crossing minimization using column generation
 ACM Journal of Experimental Algorithmics
"... Abstract. The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branchandcut approach for solving the crossing number problem has been presented in [3]. Its major drawback was the huge number of variables out of which only ..."
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Cited by 5 (4 self)
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Abstract. The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branchandcut approach for solving the crossing number problem has been presented in [3]. Its major drawback was the huge number of variables out of which only very few were actually used in the optimal solution. This restricted the algorithm to rather small graphs with low crossing number. In this paper we discuss two column generation schemes; the first is based on traditional algebraic pricing, and the second uses combinatorial arguments to decide whether and which variables need to be added. The main focus of this paper is the experimental comparison between the original approach, and these two schemes. We also compare these new results to the solutions of the best known crossing number heuristic. 1
Algorithms for the hypergraph and the minor crossing number problems
 IN PROC. ISAAC’07, VOLUME 4835 OF LNCS
, 2007
"... 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 ..."
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Cited by 5 (3 self)
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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.
NonPlanar Core Reduction of Graphs
"... 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 l ..."
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Cited by 3 (3 self)
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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.
A tighter insertionbased approximation of the crossing number
, 2011
"... Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the ..."
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Cited by 3 (2 self)
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Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the Falmostplanar graph G+F, while computing the crossing number of G+F exactly is NPhard already when F = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of Falmostplanar graphs, achieving constantfactor approximation for the large class of such graphs of bounded degrees and bounded size of F.