The structure of (t,r)-regular graphs of large order

A graph is (t,r)-regular iff it has at least one independent t-set of vertices and the open neighborhood of any such set contains exactly r vertices. Our goal is to show that when t⩾3 and the order is sufficiently large, then the structure of (t,r)-regular graphs is similar to, but not exactly the same as the structure of (2,r)-regular graphs as derived by Faudree and Knisley. That is, there is an “almost” complete kernel of order at most r surrounded by satellite cliques, all of the same order, which are “mostly” joined to the kernel.