Shrinking the Search Space for Clustered Planarity

Chimani, Markus; Klein, Karsten

doi:10.1007/978-3-642-36763-2_9

Cited by 10 publications

(6 citation statements)

References 13 publications

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“…In other words, we obtain an FPT-algorithm where the parameter is the sum of the maximum degree of the tree T and the maximum number of edges leaving a cluster. Note that this generalizes the FPT-algorithm by Chimani and Klein [7] with respect to the total number of edges connecting different clusters.…”

Section: General Clustered Graphsmentioning

confidence: 75%

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

Bläsius

Rutter

2014

Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications

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The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges.Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.

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Section: General Clustered Graphsmentioning

confidence: 75%

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

Bläsius

Rutter

2014

Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications

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“…Polynomial-time algorithms to test the c-planarity of a clustered graph C are known if C belongs to special classes of clustered graphs [7][8][9][10][11]13,15,16,18,19], including cconnected clustered graphs, that are clustered graphs C(G, T ) in which, for each cluster α, the subgraph G [α] of G induced by the vertices in α is connected [8,10,13]. Effective ILP formulations and FPT algorithms for testing c-planarity have been presented [5,6]. Generalizations of the c-planarity testing problem have also been considered [2,12].…”

Section: Introductionmentioning

confidence: 99%

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Chimani

Battista

Frati

et al. 2014

Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications

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“…Cornelsen and Wagner showed polynomiality for completely connected c-graphs, i.e., c-graphs for which not only every cluster but also the complement of each cluster is connected [17]. FPT algorithms have also been investigated [10,15]. For additional special cases, see, e.g., [2,3,4,7,14,23].A c-graph is flat when no non-trivial cluster is a subset of another, so T has only three levels: the root, the clusters, and the leaves.…”

mentioning

confidence: 99%

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Lozzo

Eppstein

Goodrich

et al. 2018

Graph-Theoretic Concepts in Computer Science

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The C-Planarity problem asks for a drawing of a clustered graph, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edgeedge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for embedded flat clustered graphs, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of embedded tree decomposition in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.A clustered graph (or c-graph) is a pair C (G, T ) with underlying graph G and inclusion tree T , i.e., a rooted tree whose leaves are the vertices of G. Each internal node µ of T represents a cluster of vertices of G (its leaf descendants) which induces a subgraph G(µ). A c-planar drawing of C (G, T ) ( Fig. 1) consists of a drawing of G and of a representation of each cluster µ as a simple closed region R(µ), i.e., a region homeomorphic to a closed disc, such that: (1) Each region R(µ) contains the drawing of G(µ). (2) For every two clusters µ, ν ∈ T , R(ν) ⊆ R(µ) if and only if ν is a descendant of µ in T . (3) No two edges cross. (4) No edge crosses any region boundary more than once. (5) No two region boundaries intersect.An interesting and challenging line of research in graph drawing concerns the computational complexity of the C-Planarity problem, which asks to test the existence of a c-planar drawing of a c-graph. This problem is notoriously difficult, particularly when (as in Fig. 1) clusters may be disconnected, faces may have unbounded size, and the cluster hierarchy may have multiple nested levels. No known subexponential-time algorithm solves the (general) C-Planarity problem, and it is unknown whether it is NP-complete, although the related problem of splitting as few clusters as possible to make a c-graph c-planar was proved NP-hard [5]. Thus, there is considerable interest in subexponential-time, slice-wise polynomial, and fixed-parameter tractable algorithms, besides polynomial-time algorithms for special cases of C-Planarity.C-Planarity was introduced by Feng, Cohen, and Eades [24], who solved it in quadratic time for the c-connected case when every cluster induces a connected subgraph. Similar results were given by Lengauer [32] using different terminology. Dahlhaus [21] claimed a linear-time algorithm for c-connected C-Planarity (with some details later provided by Cortese et al. [18]). Goodrich et al. [27] gave a cubic-time algorithm for disconnected clusters that satisfy an "extroverted" property, and Gutwenger et al.[28] provided a polynomial-time algorithm for "almost" c-connected inputs. Cornelsen and Wagne...

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Shrinking the Search Space for Clustered Planarity

Cited by 10 publications

References 13 publications

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

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