I have been reading a couple of sites regarding minors and have come across the statement that the order of deletion and contraction of edges do not matter. Why is that the case?
In fact, I came up with an example and am convinced that without additional hypotheses, this cannot be true. Perhaps I'm misunderstanding something.
Consider $G=K_4$ with one edge missing. Then we have 2 vertices of degree 3 and 2 vertices of degree 2. Let $x$ be a vertex of degree 3 and y be a vertex of degree 2 and $x,y$ are adjacent. After contracting $xy$, we get $K_3$ and after deleting an edge, we get a path of length 2 (number of edges). However, if we were to first delete the edge and then do a contraction, we will end up with $K_3$.
Am I missing something here?
To make the order of deletions and contractions not matter you must allow for multigraphs. In your case, allowing for parallel edges and loops, contracting $xy$ first results in $K_3$ with an additional edge in parallel to one of the other edges. You cannot discard that edge.