C-Planarity of C-Connected Clustered Graphs

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

doi:10.7155/jgaa.00165

Cited by 34 publications

(18 citation statements)

References 27 publications

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“…1: if G has no internal vertex then 2: 5: Γ (C) ← the drawing obtained by applying Algorithm 2 on C and Γ (C o ); 6: else 7: apply Algorithm 5 on C, obtaining either three paths P u , P v , and P z , or two paths P u and P v ; 8: for 2 ≤ i ≤ U − 1 do 9: σ (u i ) ← the unique child of σ (u i ); 10: end for 11: for 2 ≤ i ≤ V − 1 do 12: σ (v i ) ← the unique child of σ (v i ); 13: end for 14: if Algorithm 5 on C returns three paths P u = (u 1 , . .…”

Section: Input: C(g T ) and γ (C O )mentioning

confidence: 99%

Straight-Line Rectangular Drawings of Clustered Graphs

Angelini

Frati

Kaufmann

2010

Discrete Comput Geom

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Section: Input: C(g T ) and γ (C O )mentioning

confidence: 99%

Straight-Line Rectangular Drawings of Clustered Graphs

Angelini

Frati

Kaufmann

2010

Discrete Comput Geom

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“…The problem can be seen as a hierarchical variant of planarity testing; a problem for which a linear time algorithm has been known for a long time [30]. In the extensive literature devoted to c-planarity and its variants, the complexity status of only restricted special cases has been established, most notably in [2,5,17,27], see also the somewhat outdated survey [16]. The c-planarity problem is formally stated as follows.…”

Section: Introductionmentioning

confidence: 99%

Atomic Embeddability, Clustered Planarity, and Thickenability

Fulek

Tóth

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial time algorithm for this problem, thereby giving the first polynomial time algorithm for c-planarity.C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently, despite relentless efforts. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time.Our algorithm for atomic embeddability combines ideas from Carmesin's work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory.Finally we give a polynomial-time reduction from c-planarity to thickenability and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.

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“…An overview on the classical c-planarity problem can be found in [11,19,5]. Dahlhaus [12] and later Cortese et al [11] showed, that c-planarity of hierarchically clustered graphs can be solved in linear time if each cluster induces a connected subgraph. Their approaches make use of the decomposition of the graph into 3-connected components as represented by BC-and SPQR-trees.…”

Section: Introductionmentioning

confidence: 99%

Planarity of Overlapping Clusterings Including Unions of Two Partitions

Athenstädt¹,

Cornelsen²

2017

JGAA

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We show that clustered planarity with overlapping clusters as introduced by Didimo et al. [14] can be solved in polynomial time if each cluster induces a connected subgraph. It can be solved in linear time if the set of clusters is the union of two partitions of the vertex set such that, for each cluster, both the cluster and its complement, induce connected subgraphs. Clustered planarity with overlapping clusters is NP-complete, even if restricted to instances where the underlying graph is 2-connected, the set of clusters is the union of two partitions and each cluster contains at most two connected components while their complements contain at most three connected components [2].

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C-Planarity of C-Connected Clustered Graphs

Cited by 34 publications

References 27 publications

Straight-Line Rectangular Drawings of Clustered Graphs

Straight-Line Rectangular Drawings of Clustered Graphs

Atomic Embeddability, Clustered Planarity, and Thickenability

Planarity of Overlapping Clusterings Including Unions of Two Partitions

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