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11
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 14 (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.
Cplanarity of extrovert clustered graphs
 In Graph Drawing
, 2005
"... Abstract. 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 ..."
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Cited by 6 (1 self)
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Abstract. 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(n 3) time and implies an embedding algorithm with the same time complexity. 1
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 5 (1 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.
On embedding a cycle in a plane graph
 PROC. 13TH INT. SYMP. GRAPH DRAWING (GD’05), VOLUME 3843 OF LNCS
, 2006
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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 4 (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 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.
CPlanarity of CConnected Clustered Graphs  Part I – Characterization
, 2006
"... We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2 ..."
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Cited by 2 (1 self)
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We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2] we exploit such a characterization to give a linear time cplanarity testing and embedding algorithm.
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 1 (1 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.
Advances on CPlanarity Testing of Extrovert CGraphs
"... Abstract The problem of testing cplanarity of cgraphs is unknown to be NPcomplete or in P. Previous work solved this problem on some special classes of cgraphs. In particular, Goodrich, Lueker, and Sun tested cplanarity of extrovert cgraphs in O(n 3) time [5]. In this paper, we improve the ti ..."
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Abstract The problem of testing cplanarity of cgraphs is unknown to be NPcomplete or in P. Previous work solved this problem on some special classes of cgraphs. In particular, Goodrich, Lueker, and Sun tested cplanarity of extrovert cgraphs in O(n 3) time [5]. In this paper, we improve the time complexity of the testing algorithm in [5] to O(n) 2. Keywords: