  We study the lift-and-project rank of the stable set polytopes of graphs with
respect to the Lov\'{a}sz--Schrijver SDP operator $\text{LS}_+$, with a
particular focus on finding and characterizing the smallest graphs with a given
$\text{LS}_+$-rank (the least 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.
