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Todd has a “do not tamper” request up on the page at the moment, so I haven’t actually added this (and perhaps Todd intends to add precisely what I am about to say) but I think that the page needs to mention and prove the following remark: that the graph minors of $G$ are precisely the graphs $H$ such that there exists a monomorphic functor $\Pi_1 H\to\Pi_1 G$, where $\Pi_1$ is the “groupoid of paths” functor.
I haven’t thought about that. Do you have a proof in mind? Would you like to amplify on “needs to”?
I removed the “do not tamper” request.
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