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Transition Operations over Plane Trees

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LATIN 2018: Theoretical Informatics (LATIN 2018)

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Abstract

The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and geometric graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set \(\mathcal {T}(S)\) of noncrossing straight-line spanning trees on a finite point set S in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations both on general point sets and sets in convex position. In addition, we address the problem variant where operations may be performed simultaneously. We prove new lower and upper bounds for the diameters of the corresponding transition graphs and pose open problems.

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Acknowledgment

Key ideas for our results on simultaneous edge slides were discussed at the GWOP 2017 workshop in Pochtenalp, Switzerland. We thank all participants for the constructive atmosphere. Research by Nichols and Tóth was partially supported by the NSF awards CCF-1422311 and CCF-1423615. Pilz is supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.

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Authors and Affiliations

  1. California State University Northridge, Los Angeles, CA, USA

    Torrie L. Nichols & Csaba D. Tóth

  2. Department of Computer Science, ETH Zürich, Zürich, Switzerland

    Alexander Pilz & Ahad N. Zehmakan

Authors
  1. Torrie L. Nichols
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  2. Alexander Pilz
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  3. Csaba D. Tóth
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  4. Ahad N. Zehmakan
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Corresponding author

Correspondence to Alexander Pilz .

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Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Nichols, T.L., Pilz, A., Tóth, C.D., Zehmakan, A.N. (2018). Transition Operations over Plane Trees. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_60

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_60

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

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