Computing Maximum C-Planar Subgraphs

Chimani, Markus; Gutwenger, Carsten; Jansen, M. P.; Klein, Karsten; Mutzel, Petra

doi:10.1007/978-3-642-00219-9_12

Cited by 11 publications

(14 citation statements)

References 16 publications

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“…It fact, we show that c-planarity testing is fixed parameter tractable (FPT) in the number of "complicated" edges. Furthermore, our result allows a practically stronger ILP than the one described in [2].…”

Section: Contributionmentioning

confidence: 66%

“…In [2], an integer linear program (ILP) based on this idea was proposed and solved using branch-and-cut techniques. In fact, the paper considered the problem of finding a maximum c-planar clustered subgraph, allowing pure c-planarity testing as a special case with pruning-based performance optimization.…”

Section: Corollary 2 a C-graph C = (G T ) Is C-planar Iff It Can Bementioning

confidence: 99%

“…We have: ILP and Experiments. We briefly recapitulate the ILP model of [2], reduced to the special case of testing c-planarity. We have a binary variable x f for each element f ∈ F , where F denotes the set of potential augmentation edges.…”

Section: Consequences Of Theoremmentioning

confidence: 99%

“…We solve the ILP using branch-and-cut, with a separation step that checks c-/co-connectivity and planarity and adds corresponding constraints as needed, cf. [2] for details. We (re)implemented our branch-and-cut approach as a module in OGDF (www.ogdf.net) using the ABACUS framework, resulting in a more efficient code than the one used in [2], fixing also the handling of some special cases.…”

Section: Consequences Of Theoremmentioning

confidence: 99%

“…[2] for details. We (re)implemented our branch-and-cut approach as a module in OGDF (www.ogdf.net) using the ABACUS framework, resulting in a more efficient code than the one used in [2], fixing also the handling of some special cases. We experimentally compare the ILPs resulting from using the full and the shrinked search space, denoted by FULL and SHRINK, respectively, using our new consistent implementation.…”

Section: Consequences Of Theoremmentioning

confidence: 99%

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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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Section: Contributionmentioning

confidence: 66%

Section: Corollary 2 a C-graph C = (G T ) Is C-planar Iff It Can Bementioning

confidence: 99%

Section: Consequences Of Theoremmentioning

confidence: 99%

Section: Consequences Of Theoremmentioning

confidence: 99%

Section: Consequences Of Theoremmentioning

confidence: 99%

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Shrinking the Search Space for Clustered Planarity

Chimani

Klein

2013

Graph Drawing

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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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