Skeleton Chordalities
Abstract
We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) Δ skeleton-E-chordal ⇒ Δ∨ vertex-decomposable ⇒ Δ skeleton-clique-chordal. Moreover, for subflag complexes, Δ skeleton-E-chordal Longleftrightarrow Δ∨ vertex-decomposable. (For d=1 this boils down to ``G chordal Longleftrightarrow G∨ vertex-decomposable'', a result closely related to Fröberg's theorem.) (2) For subflag complexes, Δ is skeleton-E-chordal Longleftrightarrow it splits as Δ= Δ1 ∪ Δ2, with each Δi a skeleton-E-chordal induced subcomplex of Δ, and with Δ1 ∩ Δ2 a complex whose $1$-skeleton is a clique. (This generalizes ``G chordal Longleftrightarrow G splits as a union of chordal graphs that intersect in a common clique''). (3) Δ skeleton-E-chordal Longleftrightarrow every nonempty induced subcomplex of Δ has a skeleton-E-simplicial vertex. (Generalizes ``G chordal ⇔ every nonempty induced subgraph has a simplicial vertex''.) (4) Δ underclosed ⇒ Δ skeleton-weakly-chordal and weakly-closed. (Generalizes ``G interval ⇒ G chordal and co-comparability''.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)
Related
- The skeleton-blocks decomposition of Bienaymé trees, and applications to their local convergence
- The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections
- Almgren minimality of products of simplex-skeleton cones
- Parity-Skeleton First-Passage Dynamics and Optimal Intervention in Two-Sided Discrete Monitoring Systems
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BibTeX
@misc{benedetti2026skeleton,
title = {{Skeleton Chordalities}},
author = {Bruno Benedetti and Marta Pavelka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.21166},
eprint = {2607.21166},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21166},
}
RIS
TY - UNPB
TI - Skeleton Chordalities
AU - Bruno Benedetti
AU - Marta Pavelka
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2607.21166
UR - https://arxiv.org/abs/2607.21166
AB - We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) Δ skeleton-E-chordal ⇒ Δ^∨ vertex-decomposable ⇒ Δ skeleton-clique-chordal. Moreover, for subflag complexes, Δ skeleton-E-chordal Longleftrightarrow Δ^∨ vertex-decomposable. (For d=1 this boils down to ``G chordal Longleftrightarrow G^∨ vertex-decomposable'', a result closely related to Fröberg's theorem.) (2) For subflag complexes, Δ is skeleton-E-chordal Longleftrightarrow it splits as Δ= Δ_1 ∪ Δ_2, with each Δ_i a skeleton-E-chordal induced subcomplex of Δ, and with Δ_1 ∩ Δ_2 a complex whose $1$-skeleton is a clique. (This generalizes ``G chordal Longleftrightarrow G splits as a union of chordal graphs that intersect in a common clique''). (3) Δ skeleton-E-chordal Longleftrightarrow every nonempty induced subcomplex of Δ has a skeleton-E-simplicial vertex. (Generalizes ``G chordal ⇔ every nonempty induced subgraph has a simplicial vertex''.) (4) Δ underclosed ⇒ Δ skeleton-weakly-chordal and weakly-closed. (Generalizes ``G interval ⇒ G chordal and co-comparability''.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)
ER -