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Geometric Realizations with Strong Self-Duality Part I: Non-Bipartite Quadrangulations of the Projective Plane and Strongly Involutive Self-Dual Graphs — arXiv:2610.06300

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.06300
License (as archived): CC0-1.0
Archived: 2026-10-06T16:52:32.687Z
SHA-256 of the archived text: aabdddf4fcf7d4401a4719e671b01a4f43e55d2e1660bb6f1b6c75da895f1d3e
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How to cite

Geometric Realizations with Strong Self-Duality Part I: Non-Bipartite Quadrangulations of the Projective Plane and Strongly Involutive Self-Dual Graphs — arXiv:2610.06300 — arXiv (en), https://arxiv.org/abs/2610.06300 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_699c76868bd9 (Metraipolis archive dc_699c76868bd9, sha256 aabdddf4fcf7d4401a4719e671b01a4f43e55d2e1660bb6f1b6c75da895f1d3e)

BibTeX

@misc{dc699c76868bd9,
  title = {Geometric Realizations with Strong Self-Duality Part I: Non-Bipartite Quadrangulations of the Projective Plane and Strongly Involutive Self-Dual Graphs — arXiv:2610.06300},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.06300},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_699c76868bd9; sha256 aabdddf4fcf7d4401a4719e671b01a4f43e55d2e1660bb6f1b6c75da895f1d3e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.06300. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this series of articles, we study two graph classes and their various geometric representations. The first class consists of strongly involutive self-dual graphs (SISD graphs); the second consists of non-bipartite quadrangulations of the projective plane (NBQP graphs). These classes naturally arise in many geometric problems. For example, NBQP graphs are connected to diameter graphs in $\mathbb{R}^3$, tangency graphs of families of pairwise intersecting circles and pseudocircles, generalized thrackles, and graphs formed by the main diagonals of negatively self-polar polytopes. On the other hand, SISD graphs correspond to the skeletons of negatively self-polar polytopes, self-dual cones, and extremal ball-polytopes. In Part I we show a natural one-to-one correspondence between non-bipartite quadrangulations of the projective plane and 2-connected strongly involutive self-dual maps. In particular, we show that the reduced vertex-face incidence structure of a 2-connected strongly involutive self-dual map is a non-bipartite quadrangulation of the projective plane, and every non-bipartite quadrangulation of the projective plane arises that way. This allows us to translate natural properties from one class to the other. These results lay the foundation for later parts of the series.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO) ; Metric Geometry (math.MG).
Pagina del lavoro: https://arxiv.org/abs/2610.06300 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T15:19:20.022Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).