Does the Robertson-Seymour theorem apply to vertex-labeled graphs? A minor as I understand it is a graph which can be reached by a sequence of edge contractions and non-disconnecting edge deletions. It seems natural to define a label-minor in the same way, but with the restriction that an edge can be contracted only if it connects vertices of the same label. Is every label-minor-closed set of vertex-labeled graphs characterized by a finite set of minimal forbidden label-minors?
Sorry if this is a naive question as I have only recently been studying this topic, mostly on Wikipedia. Any introductory references would be appreciated. Is there a standard term for what I am calling "label-minor"?