PreprintarXiv licencearXiv
A note on generating polyhedra and quadrangulations
Abstract
A polyhedron is a planar, $3$-connected graph. We iteratively construct all polyhedra (save for pyramids) from a unique starting graph, namely the square pyramid, via two graph transformations. This builds upon a previous construction, that starts from the full class of pyramids, and applies the same transformations. In a related result, we iteratively construct all quadrangulations of the sphere where all $4$-cycles are facial, i.e., the class of radial graphs of the polyhedra (save for antibipyramids), from a unique starting graph, namely the square antibipyramid, via a unique graph transformation. This builds upon a previous construction, that starts from the full class of antibipyramids, and applies the same transformation.
Related
- A note on right-angled, ideal hyperbolic polyhedra
- Distance statistics of block-weighted planar quadrangulations
- Spanning bipartite quadrangulations in even triangulations of real projective spaces
- Geometric Realizations with Strong Self-Duality Part I: Non-Bipartite Quadrangulations of the Projective Plane and Strongly Involutive Self-Dual Graphs
- Rational decomposition of polyhedra and a conjecture of Brlek and Reutenauer
BibTeX
@misc{andreis2026note,
title = {{A note on generating polyhedra and quadrangulations}},
author = {Luisa Andreis and Riccardo W. Maffucci and Federico Polito},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20811},
eprint = {2609.20811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20811},
}
RIS
TY - UNPB
TI - A note on generating polyhedra and quadrangulations
AU - Luisa Andreis
AU - Riccardo W. Maffucci
AU - Federico Polito
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20811
UR - https://arxiv.org/abs/2609.20811
AB - A polyhedron is a planar, $3$-connected graph. We iteratively construct all polyhedra (save for pyramids) from a unique starting graph, namely the square pyramid, via two graph transformations. This builds upon a previous construction, that starts from the full class of pyramids, and applies the same transformations. In a related result, we iteratively construct all quadrangulations of the sphere where all $4$-cycles are facial, i.e., the class of radial graphs of the polyhedra (save for antibipyramids), from a unique starting graph, namely the square antibipyramid, via a unique graph transformation. This builds upon a previous construction, that starts from the full class of antibipyramids, and applies the same transformation.
ER -