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C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Authors Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, Siddharth Gupta

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Author Details

Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
David Eppstein
  • University of California, Irvine, USA
Michael T. Goodrich
  • University of California, Irvine, USA
Siddharth Gupta
  • Ben-Gurion University of the Negev, Beersheba, Israel

Funding

  • Da Lozzo, Giordano: Supported in part by H2020-MSCA-RISE project 734922 – "CONNECT", by MIUR Project "AHeAD" under PRIN 20174LF3T8, by MIUR Project "MODE" under PRIN 20157EFM5C, and by MIUR-DAAD JMP N^o 34120.
  • Eppstein, David: Supported in part by the US NSF under grants CCF-1618301 and CCF-1616248.
  • Goodrich, Michael T.: Supported in part by the US NSF under grant 1815073.
  • Gupta, Siddharth: Supported in part by the Zuckerman STEM Leadership Program and by the Frankel Foundation.

Cite As Get BibTex

Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, and Siddharth Gupta. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.IPEC.2019.9

Abstract

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA'95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs.
We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph theory
Keywords
  • Clustered planarity
  • carving-width
  • non-crossing partitions
  • FPT

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