License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.10
URN: urn:nbn:de:0030-drops-104145
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10414/
Angelini, Patrizio ;
Chaplick, Steven ;
Cornelsen, Sabine ;
Da Lozzo, Giordano ;
Roselli, Vincenzo
Morphing Contact Representations of Graphs
Abstract
We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type.
We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs.
We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set.
BibTeX - Entry
@InProceedings{angelini_et_al:LIPIcs:2019:10414,
author = {Patrizio Angelini and Steven Chaplick and Sabine Cornelsen and Giordano Da Lozzo and Vincenzo Roselli},
title = {{Morphing Contact Representations of Graphs}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {10:1--10:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-104-7},
ISSN = {1868-8969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10414},
URN = {urn:nbn:de:0030-drops-104145},
doi = {10.4230/LIPIcs.SoCG.2019.10},
annote = {Keywords: Contact representations, Triangulations, Planar morphs, Schnyder woods}
}
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Keywords: |
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Contact representations, Triangulations, Planar morphs, Schnyder woods |
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Collection: |
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35th International Symposium on Computational Geometry (SoCG 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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11.06.2019 |
