A graph property is called hereditary if it is closed with respect to deleting vertices. There are many interesting hereditary graph properties. Moreover, a number of nontrivial general facts are also known about hereditary classes of graphs, see "Global properties of hereditary classes?"
Considering complexity, hereditary graph properties include both polynomial-time decidable and NP-complete ones. We know, however, that there are a number of natural problems in NP that are candidates for NP-intermediate status, see a nice collection in Problems Between P and NPC. Among the numerous answers there, however, none of them looks like a hereditary graph property (unless I overlooked something).
Question: Do you know a hereditary graph property that is a candidate for NP-intermediate status? Or else, is there a dichotomy theorem for hereditary graph properties?