Let $G$ be a graph, and define $C_k$ to be the free abelian group on the homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge contractions. We can think of such an $H$ as having an identification of the vertices of $H$ with the vertices of $K_k$ such that all the vertices identified with $i$th vertex of $K_k$ form a connected subgraph of $H$ and there is an edge between subgraph identified with $i$ and the subgraph identified with $j$ for $i \neq j$. We can then define a map $\delta_k$ from $C_k$ to $C_{k - 1}$ by sending a homomorphism $\varphi: H \rightarrow G$ to the $\sum_{i = 1}^n (-1)^{ i + 1} \varphi|_{1 \dots i - 1, i + 1, \dots k}$ where $\varphi|_{1 \dots i - 1, i + 1, \dots k}$ represents a homomorphism from $H$ minus the subgraph identified with $i$ to $G$, which as such a graph has $K_{k - 1}$ a minor without needing to do any vertex deletions, is an element of $C_{k - 1}$ (we also reorder the vertices in the standard way. Furthermore, $\delta_{k - 1} \circ \delta_{k}$ is the zero map, so we see this forms a complex. Thus, we have a homology.
I tried to find any references to this particular definition of a homology, but was only able to find references to singular and simplicial homology. I find this particular definition interesting, as it seems like if we have a nice enough homomorphism $\psi$ from $G \rightarrow F$, and information about the homology of $F$, we can pull back the information about $F$ to say something about $G$. In particular, if $F$ is a complete graph and $\psi$ induces a surjective homomorphism from $H_n(G)$ onto $H_n(F)$, we should be able to say something about the kinds of minors $G$ has.
I would appreciate it if someone has any references to this homology, but understand if none are available.
