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Minimal Disconnected Cuts in Planar Graphs

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Fundamentals of Computation Theory (FCT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9210))

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Abstract

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity is not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected \(K_{3,3}\)-free-minor graphs and on solving a topological minor problem in the dual. We show that the latter problem can be solved in polynomial-time even on general graphs. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs.

The first author was supported by Foundation for Polish Science (HOMING PLUS/2011-4/8) and National Science Center (SONATA 2012/07/D/ST6/02432). The second author was supported by EPSRC Grant EP/K025090/1. The research of the last author was co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: ARISTEIA II.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Warsaw, Warszawa, Poland

    Marcin Kamiński

  2. School of Engineering and Computing Sciences, Durham University, Durham, UK

    Daniël Paulusma & Anthony Stewart

  3. Computer Technology Institute and Press “Diophantus”, Patras, Greece

    Dimitrios M. Thilikos

  4. Department of Mathematics, University of Athens, Athens, Greece

    Dimitrios M. Thilikos

  5. AlGCo Project-team, CNRS, LIRMM, Montpellier, France

    Dimitrios M. Thilikos

Authors
  1. Marcin Kamiński
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  2. Daniël Paulusma
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  3. Anthony Stewart
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  4. Dimitrios M. Thilikos
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Corresponding author

Correspondence to Anthony Stewart .

Editor information

Editors and Affiliations

  1. Université Paris Diderot, Paris, France

    Adrian Kosowski

  2. Université Bordeaux 1, Talence, France

    Igor Walukiewicz

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Kamiński, M., Paulusma, D., Stewart, A., Thilikos, D.M. (2015). Minimal Disconnected Cuts in Planar Graphs. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_19

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  • DOI: https://doi.org/10.1007/978-3-319-22177-9_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22176-2

  • Online ISBN: 978-3-319-22177-9

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