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Smooth weakly modular graphs — arXiv:2609.30035

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.30035
License (as archived): CC0-1.0
Archived: 2026-10-08T06:41:02.858Z
SHA-256 of the archived text: 4144583da5404d8a7a9a2cff9c4050bae459506a7f933d568b8db9ec0a2aeb42
Stable URL of this page: https://metraipolis.com/documento/dc_7fcbf3de9aca

How to cite

Smooth weakly modular graphs — arXiv:2609.30035 — arXiv (en), https://arxiv.org/abs/2609.30035 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7fcbf3de9aca (Metraipolis archive dc_7fcbf3de9aca, sha256 4144583da5404d8a7a9a2cff9c4050bae459506a7f933d568b8db9ec0a2aeb42)

BibTeX

@misc{dc7fcbf3de9aca,
  title = {Smooth weakly modular graphs — arXiv:2609.30035},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.30035},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7fcbf3de9aca; sha256 4144583da5404d8a7a9a2cff9c4050bae459506a7f933d568b8db9ec0a2aeb42},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.30035. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

A graph $G=(V,E)$ is called smooth (respectively, strongly smooth) if for any two vertices $u,v\in V$, the distance point-shadow $v|u := \{ x\in V: d(u,x)=d(u,v)+d(v,x)\}$, respectively, the point-shadow $v/u := \{ x\in V: v\in\mathrm{conv}(u,x)\}$, is geodesically convex. Smooth graphs have been introduced by Nebeský (2005) in the context of step systems. Graphs with convex point-shadows and convex distance point-shadows also naturally occur in convexity theory. Brešar et al. (2026) recently showed that several classes of graphs are smooth and that smoothness is preserved by Cartesian products, gated amalgams, and isometric subgraphs. Weakly modular graphs comprise the most important classes of graphs from Metric Graph Theory: median, modular, Helly, bridged, and dual polar graphs. In this note, we characterize smooth and strongly smooth weakly modular graphs in terms of forbidden isometric subgraphs on 5 and 7 vertices. This settles Problem 1 of the paper by Brešar et al. We also characterize prime strongly smooth weakly modular graphs, i.e., strongly smooth weakly modular graphs that cannot be obtained from smaller graphs by Cartesian products and gated amalgams.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-25 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Discrete Mathematics (cs.DM); Metric Geometry (math.MG).
Pagina del lavoro: https://arxiv.org/abs/2609.30035 — 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 2026-10-08T06:02:10.087Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-25?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).