David Wood (2023) tried to relax the Moore graph condition by asking whether there are only finitely many diameter-2 graphs with no triangle or K_2, t for any fixed t apart from stars. Let W_t be the class of such graphs. For t = 2 these are the Moore graphs of diameter 2, so the Hoffman--Singleton Theorem implies that W_2 is finite. In this talk I will show a construction of infinitely many W_3 graphs. I will also show that W_5 contains infinitely many regular graphs and that W_7 contains infinitely many Cayley graphs. This talk is based on joint work with Vladislav Taranchuk and Craig Timmons.