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The aim of this paper is to apply the framework, which was developed by Samand Snowden, to study structural properties of graph homologies, in the spiritof Ramos, Miyata and Proudfoot. Our main results concern the magnitude homologyof graphs introduced by Hepworth and Willerton. More precisely, for graphs ofbounded genus, we prove that magnitude cohomology, in each homological degree,has rank which grows at most polynomially in the number of vertices, and thatits torsion is bounded. As a consequence, we obtain analogous results for pathhomology of (undirected) graphs. We complement the work with a proof that thecategory of planar graphs of bounded genus and marked edges, with contractions,is quasi-Gr\"obner.

On finite generation in magnitude (co)homology, and its torsion