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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 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.
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:
Shrinking the Search Space for Clustered Planarity
"... Abstract. A clustered graph is a graph augmented with a hierarchical inclusion structure over its vertices, and arises very naturally in multiple application areas. While it is long known that planarity—i.e., drawability without edge crossings—of graphs can be tested in polynomial (linear) time, the ..."
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Abstract. A clustered graph is a graph augmented with a hierarchical inclusion structure over its vertices, and arises very naturally in multiple application areas. While it is long known that planarity—i.e., drawability without edge crossings—of graphs can be tested in polynomial (linear) time, the complexity for the clustered case is still unknown. In this paper, we present a new graph theoretic reduction which allows us to considerably shrink the combinatorial search space, which is of benefit for all enumerationtype algorithms. Based thereon, we give new classes of polynomially testable graphs and a practically efficient exact planarity test for general clustered graphs based on an integer linear program. 1
Clustering planarity: Small . . .
, 2009
"... We present several polynomialtime algorithms for cplanarity testing for cluster hierarchy C containing clusters of size at most three. The main result is an O(C3 + n)time algorithm for clusters of size at most three on a cycle. The result is then generalized to a special class of Eulerian graph ..."
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We present several polynomialtime algorithms for cplanarity testing for cluster hierarchy C containing clusters of size at most three. The main result is an O(C3 + n)time algorithm for clusters of size at most three on a cycle. The result is then generalized to a special class of Eulerian graphs, namely graphs obtained from a 3connected planar graph of fixed size k by multiplying and then subdividing edges. An O(3k · k · n3)time algorithm is presented. We further give an O(C2 +n)time algorithm for general 3connected planar graphs.