The Embedded Homology of Hypergraph Pairs

TL;DR

This paper extends the theory of embedded homology for hypergraphs by introducing relative homology groups, establishing key long exact sequences, and exploring potential applications in two-dimensional persistence analysis.

Contribution

It generalizes the embedded homology framework for hypergraph pairs and develops new theoretical tools like long exact and Mayer-Vietoris sequences.

Findings

Established long exact sequences for relative embedded homology.

Proved a Mayer-Vietoris sequence for hypergraph pairs.

Discussed two-dimensional persistence in this context.

Abstract

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren, and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3)(2019) 479-500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer-Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs.