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Japanese Conference on Discrete and Computational Geometry and Graphs

JCDCGG 2013: Discrete and Computational Geometry and Graphs pp 1–13Cite as

Covering Partial Cubes with Zones

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8845))

Abstract

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results.

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Notes

  1. 1.

    An arrangement is called simple if any \(d+1\) hyperplanes have empty intersection.

  2. 2.

    For a very good introduction to terminology related to partial orders and lattices we refer to Chapter 3 of Stanley, Enumerative Combinatorics Vol. I [17].

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Acknowledgments

This work was initiated at the 2nd ComPoSe Workshop held at the TU Graz, Austria, in April 2012. We thank the organizers and the participants for the great working atmosphere. We also acknowledge insightful discussions on related problems with several colleagues, in particular Michael Hoffmann (ETH Zürich) and Ferran Hurtado (UPC Barcelona).

Author information

Authors and Affiliations

  1. Université Libre de Bruxelles (ULB), Brussels, Belgium

    Jean Cardinal

  2. Technische Universität Berlin, Berlin, Germany

    Stefan Felsner

Authors
  1. Jean Cardinal
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  2. Stefan Felsner
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Corresponding authors

Correspondence to Jean Cardinal or Stefan Felsner .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications, Tokyo, Japan

    Hiro Ito

  3. Tokai University, Tokyo, Japan

    Toshinori Sakai

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Cardinal, J., Felsner, S. (2014). Covering Partial Cubes with Zones. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_1

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  • DOI: https://doi.org/10.1007/978-3-319-13287-7_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13286-0

  • Online ISBN: 978-3-319-13287-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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