PreprintCC BY 4.0arXiv
A cubical approach to homology theories for hypergraphs
Abstract
We introduce three homology theories for hypergraphs, namely Γ-homology, Box-homology, and ×-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that Box-homology is invariant under this homotopy, whereas Γ-homology and ×-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by Γ-homology that further differentiates it from Box-homology.
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BibTeX
@misc{zaidi2026cubical,
title = {{A cubical approach to homology theories for hypergraphs}},
author = {Syed Hadi Ali Zaidi and Samira Sahar Jamil},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08196},
eprint = {2610.08196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08196},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A cubical approach to homology theories for hypergraphs
AU - Syed Hadi Ali Zaidi
AU - Samira Sahar Jamil
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08196
UR - https://arxiv.org/abs/2610.08196
AB - We introduce three homology theories for hypergraphs, namely Γ-homology, Box-homology, and ×-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that Box-homology is invariant under this homotopy, whereas Γ-homology and ×-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by Γ-homology that further differentiates it from Box-homology.
ER -