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Vertex-Colored Encompassing Graphs

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Abstract

It is shown that every disconnected vertex-colored plane straight line graph with no isolated vertices can be augmented (by adding edges) into a connected plane straight line graph such that the new edges respect the coloring and the degree of every vertex increases by at most two. The upper bound for the increase of vertex degrees is best possible: there are input graphs that require the addition of two new edges incident to a vertex. The exclusion of isolated vertices is necessary: there are input graphs with isolated vertices that cannot be augmented to a connected vertex-colored plane straight line graph.

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zürich, 8092, Zurich, Switzerland

    Michael Hoffmann

  2. Department of Mathematics, California State University, Northridge, CA, 91330, USA

    Csaba D. Tóth

  3. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

Authors
  1. Michael Hoffmann
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  2. Csaba D. Tóth
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Correspondence to Csaba D. Tóth.

Additional information

Preliminary results (on encompassing trees for vertex-colored plane straight line forests) have been published in the Proceedings of the 21st ACM Symposium on Computational Geometry (Pisa, 2005), ACM Press, 2005, pp. 81–90.

M. Hoffmann is partially supported by the ESF EUROCORES programme EuroGIGA, CRP GraDR and the Swiss National Science Foundation, SNF Project 20GG21-134306.

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Hoffmann, M., Tóth, C.D. Vertex-Colored Encompassing Graphs. Graphs and Combinatorics 30, 933–947 (2014). https://doi.org/10.1007/s00373-013-1320-1

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  • DOI: https://doi.org/10.1007/s00373-013-1320-1

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