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Every Collinear Set in a Planar Graph is Free

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Abstract

We show that if a planar graph G has a plane straight-line drawing in which a subset S of its vertices are collinear, then for any set of points, X, in the plane with\(|X|=|S|\), there is a plane straight-line drawing of G in which the vertices in S are mapped to the points in X. This solves an open problem posed by Ravsky and Verbitsky (in: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, arXiv:0806.0253). In their terminology, we show that every collinear set is free. This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings.

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Notes

  1. Ravsky and Verbitsky use a somewhat stronger definition of free collinear set that we call a collinear sequence. This difference is discussed below, after the statement of Theorem 1.1.

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Acknowledgements

We are grateful to the organizers and participants for providing a stimulating research environment. The work of VD and PM was partly funded by NSERC. The work of FF was partially supported by MIUR Project “MODE” under PRIN 20157EFM5C, by MIUR Project AHeAD under PRIN 20174LF3T8, and by H2020-MSCA-RISE project 734922, “CONNECT”. The work of DG was partly funded by the ANR project GATO, under contract ANR-16-CE40-0009.

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Authors and Affiliations

  1. Department of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada

    Vida Dujmović

  2. Dipartimento di Ingegneria, Universitá Roma Tre, Rome, Italy

    Fabrizio Frati

  3. LIRMM, Université de Montpellier, CNRS, Montpellier, France

    Daniel Gonçalves

  4. School of Computer Science, Carleton University, Ottawa, Canada

    Pat Morin

  5. Institut für Informatik, Freie Universität Berlin, Berlin, Germany

    Günter Rote

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  1. Vida Dujmović
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  2. Fabrizio Frati
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  3. Daniel Gonçalves
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  4. Pat Morin
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  5. Günter Rote
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Correspondence to Pat Morin.

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Part of this research was conducted during the 5th and the 6th Workshops on Geometry and Graphs, held at the Bellairs Research Institute, March 5–10, 2017 and March 11–16, 2018. The results in this paper were presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA19) and a short version of this paper appears in the SODA19 Conference Proceedings.

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Dujmović, V., Frati, F., Gonçalves, D. et al. Every Collinear Set in a Planar Graph is Free. Discrete Comput Geom 65, 999–1027 (2021). https://doi.org/10.1007/s00454-019-00167-x

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  • DOI: https://doi.org/10.1007/s00454-019-00167-x

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