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Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

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Graph-Theoretic Concepts in Computer Science (WG 2018)

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Abstract

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 edge-edge 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.

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Notes

  1. 1.

    In Sect. 4, we exploit all the properties of the lemma. In Sect. 3, we only exploit the existence of an equivalent 2-connected c-graph with maximum face size \(\kappa =O(\ell )\).

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Acknowledgments

Supported in part by MIUR Project “MODE” under PRIN 20157EFM5C, by H2020-MSCA-RISE project 734922 - “CONNECT”, by MIUR-DAAD JMP N\(^\circ \) 34120, and by NSF grants CCF-1618301 and CCF-1616248. This article also reports on work supported by the U.S. Defense Advanced Research Projects Agency (DARPA) under agreement no. AFRL FA8750-15-2-0092. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

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Authors and Affiliations

  1. Roma Tre University, Rome, Italy

    Giordano Da Lozzo

  2. University of California, Irvine, USA

    David Eppstein, Michael T. Goodrich & Siddharth Gupta

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  1. Giordano Da Lozzo
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  2. David Eppstein
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  3. Michael T. Goodrich
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  4. Siddharth Gupta
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Correspondence to Giordano Da Lozzo .

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Editors and Affiliations

  1. Universität Rostock, Rostock, Germany

    Andreas Brandstädt

  2. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Ekkehard Köhler

  3. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Klaus Meer

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Da Lozzo, G., Eppstein, D., Goodrich, M.T., Gupta, S. (2018). Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_10

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  • DOI: https://doi.org/10.1007/978-3-030-00256-5_10

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