The reachability homology of a directed graph
Abstract
The last decade has seen the development of path homology and magnitude homology -- two homology theories of directed graphs, each satisfying classic properties such as Kunneth and Mayer-Vietoris theorems. Recent work of Asao has shown that magnitude homology and path homology are related, appearing in different pages of a certain spectral sequence. Here we study the target of that spectral sequence, which we call reachability homology. We prove that it satisfies appropriate homotopy invariance, Kunneth, excision, and Mayer-Vietoris theorems, these all being stronger than the corresponding properties for either magnitude or path homology.
Keywords
Cite
@article{arxiv.2312.01378,
title = {The reachability homology of a directed graph},
author = {Richard Hepworth and Emily Roff},
journal= {arXiv preprint arXiv:2312.01378},
year = {2025}
}
Comments
20 pages. Version 2: acknowledgements and references updated; a few typos corrected; minor improvements to the exposition. Version 3: references updated. This is the final version, published in International Mathematics Research Notices