# Morphing Contact Representations of Graphs

Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, Vincenzo Roselli

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingAcademicpeer-review

## 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.

Original language English 35th International Symposium on Computational Geometry (SoCG 2019) Gill Barequet, Yusu Wang Dagstuhl, Germany Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 10:1-10:16 129 978-3-95977-104-7 https://doi.org/10.4230/LIPICS.SOCG.2019.10 Published - 2019 Yes

### Publication series

Series Leibniz International Proceedings in Informatics (LIPIcs) 129 1868-8969
• ### Morphing Triangle Contact Representations of Triangulations

Angelini, P., Chaplick, S., Cornelsen, S., Da Lozzo, G. & Roselli, V., Mar 2023, 70, 3, p. 991-1024 34 p.