2026-08-17T05:05:51Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10414 2024-03-06T10:45:55Z ddc:004 open_access

Morphing Contact Representations of Graphs Angelini, Patrizio Chaplick, Steven Cornelsen, Sabine Da Lozzo, Giordano Roselli, Vincenzo Contact representations Triangulations Planar morphs Schnyder woods 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Patrizio Angelini and Steven Chaplick and Sabine Cornelsen and Giordano Da Lozzo and Vincenzo Roselli 2019 Is Part Of LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2019.10 urn:nbn:de:0030-drops-104145 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.10 eng https://creativecommons.org/licenses/by/3.0/legalcode