@article{f172ccb0f29e404bba077b4d073246f5,
title = "Morphing Triangle Contact Representations of Triangulations",
abstract = "A morph is a continuous transformation between two representations of a graph. We consider the problem of morphing between contact representations of a plane graph. In an F-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family F 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 G. In a morph between two F-contact representations we insist that at each time step (continuously throughout the morph) we have an F contact representation. We focus on the case when F is the family of triangles in R-2 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. Moreover, they naturally correspond to 3-orientations. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We characterize the pairs of RT-representations admitting a morph between each other via the respective 3-orientations. Our characterization leads to a polynomial-time algorithm to decide whether there is a morph between two RT-representations of an n-vertex plane triangulation, and, if so, computes a morph with O(n(2)) steps. Each of these steps is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. Our characterization also implies 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.",
author = "Patrizio Angelini and Steven Chaplick and Sabine Cornelsen and {Da Lozzo}, Giordano and Vincenzo Roselli",
note = "Funding Information: This research began at the Graph and Network Visualization Workshop 2018 (GNV{\textquoteright}18) in Heiligkreuztal. Our work was supported in part by the German Research Foundation DFG grant WO 758/11-1 (Chaplick), by DFG - Project-ID 50974019 - TRR 161 (B06) (Cornelsen), by MIUR Project “AHeAD” under PRIN 20174LF3T8, and by H2020-MSCA-RISE project 734922 - “CONNECT” (Da Lozzo and Roselli). Publisher Copyright: {\textcopyright} 2023, The Author(s).",
year = "2023",
month = mar,
doi = "10.1007/s00454-022-00475-9",
language = "English",
volume = "70",
pages = "991--1024",
journal = "Discrete & Computational Geometry",
issn = "0179-5376",
publisher = "Springer Verlag",
number = "3",
}