Year: 2026
Type: preprint
Abstract: We introduce three homology theories for hypergraphs, namely $Γ$-homology, $\Box$-homology, and $\times$-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 $\times$-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.
Source: arXiv (Cornell University)
Authors: Zaidi, Syed Hadi Ali, Jamil, Samira Sahar
Institution: Institute of Business Administration Karachi
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220981604