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 lowerright half of axisparallel rectangles. Such RTrepresentations 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 straightline trajectories. We provide a polynomialtime algorithm that decides whether there is a piecewise linear morph between two RTrepresentations 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 4connected plane triangulations there is a morph between every pair of RTrepresentations where the "topmost" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RTrepresentations of any 4connected 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 = {10:110:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771047},
ISSN = {18688969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10414},
URN = {urn:nbn:de:0030drops104145},
doi = {10.4230/LIPIcs.SoCG.2019.10},
annote = {Keywords: Contact representations, Triangulations, Planar morphs, Schnyder woods}
}
Keywords: 

Contact representations, Triangulations, Planar morphs, Schnyder woods 
Collection: 

35th International Symposium on Computational Geometry (SoCG 2019) 
Issue Date: 

2019 
Date of publication: 

11.06.2019 