We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász--Schrijver SDP operator $\text{LS}_+$, with a particular focus on finding and characterizing the smallest graphs with a given $\text{LS}_+$-rank (the needed number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We introduce a generalized vertex-stretching operation that appears to be promising in generating $\text{LS}_+$-minimal graphs and study its properties. We also provide several new $\text{LS}_+$-minimal graphs, most notably the first known instances of $12$-vertex graphs with $\text{LS}_+$-rank $4$, which provides the first advance in this direction since Escalante, Montelar, and Nasini's discovery of a $9$-vertex graph with $\text{LS}_+$-rank $3$ in 2006.