On graphs with few disjoint t-star minors

For fixed positive integers t≥3 and k, consider the class of graphs which have at most k disjoint minors isomorphic to a t-star. We shall see that almost all of these graphs contain k 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.