The triangular extensions of a generalized quadrangle of order \((3,3)\) (Q1909613)

A triangular extended generalized quadrangle of order \((s,t)\) can be defined as a graph \(\Gamma\) such that for each vertex \(u\) of \(\Gamma\) the subgraph \(\Gamma (u)\) induced on the neighbourhood of \(u\) is isomorphic to the collinearity graph of a generalized quadrangle of order \((s,t)\). It is known that if \(u\) and \(v\) are vertices at distance two then \(\Gamma (u) \cap \Gamma (v)\) is a hyperoval in \(\Gamma (u)\). The author determines all hyperovals in the two generalized quadrangles of order (3,3) by a computer search and uses this classification to show that there is precisely one triangular extended generalized quadrangle of order (3,3).