User blog comment:Wythagoras/Extension to Friedmann's stuff/@comment-11227630-20150501095933/@comment-11227630-20150501111642

In your question, "contracting some edges" merges 2 labelled vertices into one vertex with the "smaller" label (or I misunderstand?), then this graph minor is still a well-quasi-order.

In this question, "contracting some edges" merges 2 labelled vertices into one vertex only when they have the same label, then this graph minor is not a well-quasi-order.

Here I've a thought: "contracting some edges" merges 2 labelled vertices (one have label a and another have label b) into one vertex that has label a or b. Is that a well-quasi-order?

However, well-quasi-order doesn't means the existance of labelled SCG(a,b). A graph sequence with no graph minor between Gi and Gj (for all i<j) cannot be infinite, but it can be any finite long; if so, "the longest sequence" will not exist.