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11
Radial Level Planarity Testing and Embedding in Linear Time
 Journal of Graph Algorithms and Applications
, 2005
"... A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines an ..."
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Cited by 19 (9 self)
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A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are level nonplanar biconnected components. Our main results are linear time algorithms for radial level planarity testing and for computing an embedding. We introduce PQRtrees as a new data structure where Rnodes and associated templates for their manipulation are introduced to deal with rings. Our algorithms extend level planarity testing and embedding algorithms which use PQtrees.
An SDP Approach to Multilevel Crossing Minimization
"... We present an approach based on semidefinite programs (SDP) to tackle the multilevel crossing minimization problem. Thereby, we are given a layered graph (i.e., the graph’s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when drawin ..."
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Cited by 7 (5 self)
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We present an approach based on semidefinite programs (SDP) to tackle the multilevel crossing minimization problem. Thereby, we are given a layered graph (i.e., the graph’s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the socalled Sugiyama framework. The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed
Optimal klevel planarization and crossing minimization
 In Proceedings of the Symposium on Graph Drawing [GD’10], LNCS
, 2010
"... Abstract. An important step in laying out hierarchical network diagrams is to order the nodes on each level. The usual approach is to minimize the number of edge crossings. This problem is NPhard even for two layers when the first layer is fixed. Hence, in practice crossing minimization is performe ..."
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Cited by 5 (0 self)
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Abstract. An important step in laying out hierarchical network diagrams is to order the nodes on each level. The usual approach is to minimize the number of edge crossings. This problem is NPhard even for two layers when the first layer is fixed. Hence, in practice crossing minimization is performed using heuristics. Another suggested approach is to maximize the planar subgraph, i.e. find the least number of edges to delete to make the graph planar. Again this is performed using heuristics since minimal edge deletion for planarity is NPhard. We show that using modern SAT and MIP solving approaches we can find optimal orderings for minimal crossing or minimal edge deletion for planarization on reasonably sized graphs. These exact approaches provide a benchmark for measuring quality of heuristic crossing minimization and planarization algorithms. Furthermore, we can straightforwardly extend our approach to minimize crossings followed by maximizing planar subgraph or vice versa; these hybrid approaches produce noticeably better layout then either crossing minimization or planarization alone. 1
Track Planarity Testing and Embedding
 PROC. SOFTWARE SEMINAR: THEORY AND PRACTICE OF INFORMATICS, SOFSEM 2004
, 2004
"... A track graph is a graph with its vertex set partitioned into horizontal levels. It is track planar if there are permutations of the vertices on each level such that all edges can be drawn as weak monotone curves without crossings. The novelty and generalisation over level planar graphs is that ..."
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Cited by 4 (3 self)
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A track graph is a graph with its vertex set partitioned into horizontal levels. It is track planar if there are permutations of the vertices on each level such that all edges can be drawn as weak monotone curves without crossings. The novelty and generalisation over level planar graphs is that horizontal edges connecting consecutive vertices on the same level are allowed. We show that track planarity can be reduced to level planarity in linear time. Hence, there are time algorithms for the track planarity test and for the computation of a track planar embedding.
Clustered Level Planarity
 Proc. 30th Int. Conf. Current Trends in Theory and Practice of Computer Science (SOFSEM’04
, 2004
"... Planarity is an important concept in graph drawing. It is generally accepted that planar drawings are well understandable. Recently, several variations of planarity have been studied for advanced graph concepts such as klevel graphs and clustered graphs. In klevel graphs, the vertices are partitio ..."
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Cited by 3 (1 self)
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Planarity is an important concept in graph drawing. It is generally accepted that planar drawings are well understandable. Recently, several variations of planarity have been studied for advanced graph concepts such as klevel graphs and clustered graphs. In klevel graphs, the vertices are partitioned into k levels and the vertices of one level are drawn on a horizontal line. In clustered graphs, there is a recursive clustering of the vertices according to a given nesting relation. In this paper we combine the concepts of level planarity and clustering and introduce clustered klevel graphs. For connected clustered level graphs we show that clustered klevel planarity can be tested in O(kV) time.
Data Visualization Through Graph Drawing
 Comput. Statist
, 2001
"... . In this paper the problem of visualizing categorical multivariate data sets is considered. By representing the data as the adjacency matrix of an appropriately defined bipartite graph, the problem is transformed to one of graph drawing. A general graph drawing framework is introduced, the corr ..."
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Cited by 2 (0 self)
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. In this paper the problem of visualizing categorical multivariate data sets is considered. By representing the data as the adjacency matrix of an appropriately defined bipartite graph, the problem is transformed to one of graph drawing. A general graph drawing framework is introduced, the corresponding mathematical problem defined and an algorithmic approach for solving the necessary optimization problem discussed. The new approach is illustrated through several examples. 1.
Exact Approaches to Multilevel Vertical Orderings
, 2011
"... We present a semidefinite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This multilevel vertical ordering (MLVO) problem falls into the class of quadratic ordering problems. It is conceptually r ..."
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Cited by 2 (2 self)
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We present a semidefinite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This multilevel vertical ordering (MLVO) problem falls into the class of quadratic ordering problems. It is conceptually related to the wellstudied problem of multilevel crossing minimization (MLCM), but offers certain interesting novel properties: we not only have to consider the pure relative ordering of the nodes, but their final absolute ranks (i.e., positions) within the ordered levels. Furthermore, MLVO is a natural quadratic problem that does not only consist of multiple sequentially linked bilevel quadratic ordering problems, but is a genuine multilevel quadratic ordering problem. This allows us to describe the graphs’ structures more compactly and therefore obtain (near)optimal, (well)readable drawings of graphs too large for MLCM. We show (theoretically and experimentally) that these properties lead to the situation that approaches based on ILPs and QPs are inapplicable, even for small sparse graphs, while the SDP works surprisingly well in practice. This is in stark contrast to other ordering problems as, e.g., MLCM, where such graphs are typically solved more efficiently with ILPs. In this paper we present a motivation, mathematical models, strengthening constraints for ILPs and QPs, and an SDP relaxation for MLVO. We compare the relevant models from the polyhedral point of view, and conduct a series of experiments (including a comparison to MLCM) to showcase our SDP’s applicability. We conclude with sketching further applications in scheduling and ranking problems, where MLVO occurs apart from graph drawing. Key words: Quadratic ordering problem, ILP and SDP approaches, multilayer graph drawings, crossing minimization. History:... 1
Multilevel verticality optimization: Concept, strategies, and drawing scheme
, 2011
"... Abstract. In traditional multilevel graph drawing—known as Sugiyama’s framework—the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative m ..."
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Cited by 1 (1 self)
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Abstract. In traditional multilevel graph drawing—known as Sugiyama’s framework—the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and show that drawings that are good w.r.t. verticality in fact also have a low number of crossings. We present heuristic and exact approaches to tackle the verticality problem and study them in practice. Furthermore, we present a new drawing scheme (inherently bundling edges and drawing them monotonously), especially suitable for verticality optimization. It works without the traditional subdivision of edges, i.e., edges may span multiple levels, and therefore potentially allows to tackle larger graphs. 1
Graph Simultaneous Embedding Tool, GraphSET
"... Problems in simultaneous graph drawing involve the layout of several graphs on a shared vertex set. This paper describes a Graph Simultaneous Embedding Tool, GraphSET, designed to allow the investigation of a wide range of embedding problems. GraphSET can be used in the study of several variants of ..."
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Problems in simultaneous graph drawing involve the layout of several graphs on a shared vertex set. This paper describes a Graph Simultaneous Embedding Tool, GraphSET, designed to allow the investigation of a wide range of embedding problems. GraphSET can be used in the study of several variants of simultaneous embedding including simultaneous geometric embedding, simultaneous embedding with fixed edges and colored simultaneous embedding with the vertex set partitioned into color classes. The tool has two primary uses: (i) studying theoretical problems in simultaneous graph drawing through the production of examples and counterexamples and (ii) producing layouts of given classes of graphs using builtin implementations of known algorithms. GraphSET along with movies illustrating its utility are available at
Communicated by:
, 2013
"... In traditional multilevel graph drawing—known as Sugiyama’s framework—the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and ..."
Abstract
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In traditional multilevel graph drawing—known as Sugiyama’s framework—the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and show that drawings that are good w.r.t. verticality in fact also have a low number of crossings. We present heuristic and exact approaches to tackle the verticality problem and study them in practice. Furthermore, we present a new drawing scheme (inherently bundling edges and drawing them monotonously), especially suitable for verticality optimization. It works without the traditional subdivision of edges, i.e., edges may span multiple levels, and therefore potentially allows to tackle larger graphs. Submitted: