Clustering Cycles into Cycles of Clusters

Cortese, Pier Francesco; Battista, Giuseppe Di; Patrignani, Maurizio; Pizzonia, Maurizio

doi:10.1007/978-3-540-31843-9_12

Cited by 16 publications

(28 citation statements)

References 10 publications

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“…Gutwenger et al [13] solved in O(n 2 ) time the case of almost c-connected c-graphs, namely, those cgraphs in which either each disconnected cluster ν ∈ τ has its parent and all siblings connected, or all disconnected clusters lie on a path in τ . Cortese et al [4] recently solved in polynomial time another special case, which we call the cycles of clusters, where the underlying graph is a cycle and the clusters at each level of the inclusion tree, when contracted into vertices, also form a cycle. To the best of our knowledge, these three classes of c-graphs are the only ones for which c-planarity has been tested in polynomial time.…”

Section: Previous Resultsmentioning

confidence: 99%

“…Extrovert c-graphs appear to allow a greater degree of disconnectivity than almost-connected c-graphs, since many sibling clusters are allowed to be disconnected. Extrovert c-graphs are also more flexible than the cycles of clusters of [4].…”

Section: Extrovert C-graphsmentioning

confidence: 99%

“…Previous work provides effective tests only for a few special classes of c-graphs [4,5,12,13]. In this paper we define and test a new class of c-graphs, which generalizes the result in [5,12] but is not comparable with [4,13].…”

Section: Introductionmentioning

confidence: 99%

“…Depending on where we start, and whether we read clockwise or counterclockwise, we can obtain various permutations; we will say these permutations are circularly equivalent. For example, the permutations (3, 5, 2, 4, 1, 6), (2,4,1,6,3,5), and (5, 3, 6, 1, 4, 2) are circularly equivalent. We call an equivalence class of this relation a circular permutation.…”

Section: Circular Permutationsmentioning

confidence: 99%

“…There is no crossing between boundary curves of any two clusters. 4. There is exactly one crossing between an edge (x, y) and a boundary curve b(ν) if x ∈ ν and y ∈ ν.…”

Section: Introductionmentioning

confidence: 99%

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C-Planarity of Extrovert Clustered Graphs

2006

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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 c-planarity 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 NP-complete or in P. In this paper, we show how to solve the c-planarity 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.

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Section: Previous Resultsmentioning

confidence: 99%

Section: Extrovert C-graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Circular Permutationsmentioning

confidence: 99%

“…There is no crossing between boundary curves of any two clusters. 4. There is exactly one crossing between an edge (x, y) and a boundary curve b(ν) if x ∈ ν and y ∈ ν.…”

Section: Introductionmentioning

confidence: 99%

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C-Planarity of Extrovert Clustered Graphs

2006

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On Embedding a Cycle in a Plane Graph

et al. 2006

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Consider a planar drawing Γ of a planar graph G such that the vertices are drawn as small circles and the edges are drawn as thin strips. Consider a cycle c of G. Is it possible to draw c as a nonintersecting closed curve inside Γ, following the circles that correspond in Γ to the vertices of c and the strips that connect them? We show that this test can be done in polynomial time and study this problem in the framework of clustered planarity for highly non-connected clustered graphs.

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Clustered Planarity = Flat Clustered Planarity

Cortese

Patrignani

2018

Lecture Notes in Computer Science

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The complexity of deciding whether a clustered graph admits a clustered planar drawing is a long-standing open problem in the graph drawing research area. Several research efforts focus on a restricted version of this problem where the hierarchy of the clusters is 'flat', i.e., no cluster different from the root contains other clusters. We prove that this restricted problem, that we call Flat Clustered Planarity, retains the same complexity of the general Clustered Planarity problem, where the clusters are allowed to form arbitrary hierarchies. We strengthen this result by showing that Flat Clustered Planarity is polynomial-time equivalent to Independent Flat Clustered Planarity, where each cluster induces an independent set. We discuss the consequences of these results.

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Clustering Cycles into Cycles of Clusters

Cited by 16 publications

References 10 publications

C-Planarity of Extrovert Clustered Graphs

C-Planarity of Extrovert Clustered Graphs

On Embedding a Cycle in a Plane Graph

Clustered Planarity = Flat Clustered Planarity

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