$\mathrm{\Delta }$-Y Moves and Partial Steiner Systems

We study graph systems generated by $\mathrm{\Delta }$-Y and Y-$\mathrm{\Delta }$ moves and prove that, for initial graphs of minimum degree at least $7$, $\mathrm{\Delta }$-Y moves alone generate every reachable graph. As a consequence, for $n\ge 8$, the $\mathrm{\Delta }$-Y system generated from the complete graph of $n$ vertices is canonically isomorphic as graded directed graphs to the space of partial Steiner systems on $n$ points.