The Proof of Wagner's Conjecture and Why It Matters
medium.com
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Is there a cubic algorithm for determining one graph is a minor of another? Last I checked it was an existential proof and no one actually had one
This paper: http://kam.mff.cuni.cz/~fiala/tw.pdf gives a deeper (but still very comprehensible) overview look at corollary 2.5 and for more specifics.