  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.
