For fixed positive integers and , consider the class of graphs which have at most disjoint minors isomorphic to a -star. We shall see that almost all of these graphs contain vertices such that deleting them leaves a graph with no such minor. This holds for both labelled and unlabelled graphs, and answers a question of Bernardi, Noy and Welsh. We also estimate the asymptotic proportion of graphs in the class which do not have this property.