{"ID":23664231,"CreatedAt":"2026-09-18T08:14:42.696445972Z","UpdatedAt":"2026-09-18T08:14:42.696445972Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20811","arxiv_id":"2609.20811","title":"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.","short_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. I...","url_abs":"https://arxiv.org/abs/2609.20811","url_pdf":"https://arxiv.org/pdf/2609.20811v1","authors":"[\"Luisa Andreis\",\"Riccardo W. Maffucci\",\"Federico Polito\"]","published":"2026-09-17T17:58:54Z","proceeding":"math.CO","tasks":"[\"math.CO\"]","methods":"[]","has_code":false}