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16
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 71 (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 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 18 (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.
Clustering Cycles into Cycles of Clusters
, 2005
"... In this paper we study simple families of clustered graphs that are highly non connected. We start by studying 3cluster cycles, that are clustered graphs such that the underlying graph is a simple cycle and there are three clusters all at the same level. We show that in this case testing the cplan ..."
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Cited by 18 (7 self)
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In this paper we study simple families of clustered graphs that are highly non connected. We start by studying 3cluster cycles, that are clustered graphs such that the underlying graph is a simple cycle and there are three clusters all at the same level. We show that in this case testing the cplanarity can be done efficiently and give an efficient drawing algorithm. Also, we characterize 3cluster cycles in terms of formal grammars. Finally, we generalize the results on 3cluster cycles considering clustered graphs that have a cycle structure at each level of the inclusion tree. Even in this case we show efficient cplanarity testing and drawing algorithms.
Completely connected clustered graphs
 IN PROC. 29TH INTL. WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE (WG 2003), VOLUME 2880 OF LNCS
, 2003
"... Planar drawings of clustered graphs are considered. We introduce the notion of completely connected clustered graphs, i.e. hierarchically clustered graphs that have the property that not only every cluster but also each complement of a cluster induces a connected subgraph. As a main result, we prove ..."
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Cited by 17 (1 self)
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Planar drawings of clustered graphs are considered. We introduce the notion of completely connected clustered graphs, i.e. hierarchically clustered graphs that have the property that not only every cluster but also each complement of a cluster induces a connected subgraph. As a main result, we prove that a completely connected clustered graph is cplanar if and only if the underlying graph is planar. Further, we investigate the influence of the root of the inclusion tree to the choice of the outer face of the underlying graph and vice versa.
Efficient cplanarity testing for embedded flat clustered graphs with small faces
 PROC. GRAPH DRAWING, VOLUME 4875 OF LNCS
, 2008
"... Let C be a clustered graph and suppose that the planar embedding of its underlying graph is fixed. Is testing the cplanarity of C easier than in the variable embedding setting? In this paper we give a first contribution towards answering the above question. Namely, we characterize cplanar embedded ..."
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Cited by 17 (3 self)
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Let C be a clustered graph and suppose that the planar embedding of its underlying graph is fixed. Is testing the cplanarity of C easier than in the variable embedding setting? In this paper we give a first contribution towards answering the above question. Namely, we characterize cplanar embedded flat clustered graphs with at most five vertices per face and give an efficient testing algorithm for such graphs. The results are based on a more general methodology that sheds new light on the cplanarity testing problem.
Cplanarity of extrovert clustered graphs
 IN GRAPH DRAWING
, 2005
"... A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The cplanarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions and wi ..."
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Cited by 9 (1 self)
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A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The cplanarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions and with each edge (drawn as a curve between vertex points) crossing the boundary of each region at most once. Unfortunately, as with the graph isomorphism problem, it is open as to whether the cplanarity problem is NPcomplete or in P. In this paper, we show how to solve the cplanarity problem in polynomial time for a new class of clustered graphs, which we call extrovert clustered graphs. This class is quite natural (we argue that it captures many clustering relationships that are likely to arise in practice) and includes the clustered graphs tested in previous work by Dahlhaus, as well as Feng, Eades, and Cohen. Interestingly, this class of graphs does not include, nor is it included by, a class studied recently by Gutwenger et al.; therefore, this paper offers an alternative advancement in our understanding of the efficient drawability of clustered graphs in a planar way. Our testing algorithm runs in O(n³) time and implies an embedding algorithm with the same time complexity.
Twopage book embedding and clustered graph planarity
, 2009
"... Abstract: A 2page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2page book embedding ..."
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Cited by 7 (0 self)
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Abstract: A 2page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2page book embedding is known to be NPcomplete. In this paper, we study the problem of testing whether a given graph admits a 2page book embedding with a fixed edge partition. Based on structural properties of planar graphs, we prove that the problem of testing and finding a 2page book embedding of a graph with a partitioned edge set can be solved in linear time. As an application of our main result, we consider the problem of testing planarity of clustered graphs. The complexity of testing clustered graph planarity is a long standing open problem in Graph Drawing. Recently, polynomial time algorithms have been found for several classes of clustered graphs in which both the structure of the underlying graphs and clustering structure are restricted. However, when the underlying graph is disconnected, the problem remains open. Our book embedding results imply that the clustered planarity problem can be solved in linear time for a certain class of clustered graphs with arbitrary underlying graphs (i.e. possibly disconnected). 1
CPlanarity of cconnected clustered graphs
, 2008
"... We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we ..."
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Cited by 6 (4 self)
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We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a lineartime cplanarity testing and embedding algorithm for cconnected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of wellknown data structures as SPQRtrees and BCtrees. If the test fails, the algorithm identifies a structural element responsible for the noncplanarity of the input clustered graph.
Overlapping Cluster Planarity
, 2008
"... This paper investigates a new direction in the area of cluster planarity by addressing the following question: Let G be a graph along with a hierarchy of vertex clusters, where clusters can partially intersect. Does G admit a drawing where each cluster is inside a simple closed region, no two edges ..."
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Cited by 5 (0 self)
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This paper investigates a new direction in the area of cluster planarity by addressing the following question: Let G be a graph along with a hierarchy of vertex clusters, where clusters can partially intersect. Does G admit a drawing where each cluster is inside a simple closed region, no two edges intersect, and no edge intersects a region twice? We investigate the interplay between this problem and the classical cluster planarity testing problem where clusters are not allowed to partially intersect. Characterizations, models, and algorithms are discussed.
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 4 (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.