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URN: urn:nbn:de:0030-drops-104145
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### Morphing Contact Representations of Graphs

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### 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}
}
```

 Keywords: Contact representations, Triangulations, Planar morphs, Schnyder woods Seminar: 35th International Symposium on Computational Geometry (SoCG 2019) Issue date: 2019 Date of publication: 11.06.2019

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