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Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

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Graph-Theoretic Concepts in Computer Science (WG 2014)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 8747))

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Abstract

Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.

On the positive side we give two polynomial time algorithms for the grid contact case. The first one is for the case when all vertical segments are pre-represented (note: the problem remains NP-complete when a subset of the vertical segments is specified, even if none of the horizontals are). Secondly, we show that the case when the vertical segments have only their \(x\)-coordinates specified (i.e., they are ordered horizontally) is polynomially equivalent to level planarity, which is known to be solvable in polynomial time.

S. Chaplick: Supported by ESF Eurogiga project GraDR.

J. Kratochvíl: Supported by GAČR project GA14-14179S and partially by ESF Eurogiga project GraDR as GAČR GIG/11/E023.

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Notes

  1. 1.

    The only exception is formed by partial subtree-in-tree or path-in-tree representations of chordal graphs, but there the NP-hardness follows from limited space issues, not any geometrical ones [18].

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Author information

Authors and Affiliations

  1. Institut Für Mathematik, Technische Universität Berlin, Berlin, Germany

    Steven Chaplick

  2. University of Bordeaux, CNRS - LaBRI, UMR 5800, 33400, Talence, France

    Paul Dorbec

  3. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic

    Jan Kratochvíl

  4. Université Montpellier 2, CNRS - LIRMM, Montpellier, France

    Mickael Montassier

  5. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, USA

    Juraj Stacho

Authors
  1. Steven Chaplick
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  2. Paul Dorbec
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  3. Jan Kratochvíl
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  4. Mickael Montassier
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  5. Juraj Stacho
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Correspondence to Steven Chaplick .

Editor information

Editors and Affiliations

  1. LITA, Université de Lorraine, Metz Cedex 01, France

    Dieter Kratsch

  2. Université d'Orléans, Orléans, France

    Ioan Todinca

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Chaplick, S., Dorbec, P., Kratochvíl, J., Montassier, M., Stacho, J. (2014). Contact Representations of Planar Graphs: Extending a Partial Representation is Hard. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_12

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  • DOI: https://doi.org/10.1007/978-3-319-12340-0_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12339-4

  • Online ISBN: 978-3-319-12340-0

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