Clustering Cycles into Cycles of Clusters

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

doi:10.7155/jgaa.00115

Cited by 34 publications

(23 citation statements)

References 11 publications

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“…• cycles of clusters, in which the hierarchy is flat, the underlying graph is a simple cycle, and the clusters are arranged in a cycle [5]; the clustering hierarchy is flat if all clusters, but for the root, are at the same level.…”

Section: Introductionmentioning

confidence: 99%

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Battista¹,

Frati²

2009

JGAA

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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 c-planarity 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 c-planar 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 c-planarity testing problem.

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Section: Introductionmentioning

confidence: 99%

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Battista¹,

Frati²

2009

JGAA

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“…Namely, we perform a reduction from NAE3SAT [9], which takes in input a collection of clauses, each consisting of three literals, and asks whether a truth assignment to the variables exists such that each clause has at least one true literal and at least one false literal. Given a clustered graph C(G, T ) and a vertex v of G with four incident edges e 1 , e 2 , e 3 , and e 4 , we introduce a gadget that forces such edges to appear in this circular order around v in any c-planar drawing of any clustered graph obtained from C with less than σ splits. We construct around v a pinwheel gadget of size σ by inserting, in each edge e i , 2σ vertices v i,j , with j = 1, .…”

Section: Non-c-connected Clustered Graphsmentioning

confidence: 99%

“…We refer to [5] for basic definitions about graphs and embeddings, and to [8,4,11,3,10,1,6,2,13,12] for basic definitions about clustered graphs and c-planar drawings.…”

Section: Introductionmentioning

confidence: 99%

“…Restrictions on the c-planarity testing problem that have been considered in the literature include: (i) assuming that each cluster induces a small number of connected components [8,4,11,10,1,2,12] (in particular, the case in which the graph is c-connected, that is, each cluster induces one connected component, has been deeply investigated); (ii) considering only flat hierarchies, where all clusters different from the root of T are children of the root [3,6]; (iii) focusing on particular families of underlying graphs [3,13]; and (iv) fixing the embedding of the underlying graph [6,12].…”

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confidence: 99%

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Splitting Clusters to Get C-Planarity

2010

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Abstract. In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the SPLIT-C-PLANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that SPLIT-C-PLANARITY is NP-complete for c-connected clustered triangulations and for non-c-connected clustered paths and cycles. On the other hand, we present a polynomial-time algorithm for flat c-connected clustered graphs whose underlying graph is a biconnected seriesparallel graph, both in the fixed and in the variable embedding setting, when the splits are assumed to maintain the c-connectivity of the clusters.

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“…Currently only some special graph classes are known to be solvable in polynomial time, e.g., [4,7,8,[10][11][12]. Most importantly, there is a linear-time algorithm to check cluster-connected graphs (i.e., G [σ] is connected for each σ) [5].…”

Section: Introductionmentioning

confidence: 99%

Shrinking the Search Space for Clustered Planarity

Chimani

Klein

2013

Graph Drawing

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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 crossingsof 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 enumeration-type 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.

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Clustering Cycles into Cycles of Clusters

Cited by 34 publications

References 11 publications

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Splitting Clusters to Get C-Planarity

Shrinking the Search Space for Clustered Planarity

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