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.

翻译：我们研究图的稳定集多面体关于Lovász–Schrijver SDP算子$\text{LS}_+$的lift-and-project秩，特别关注寻找并刻画具有给定$\text{LS}_+$-秩（即从分数稳定集多面体出发，通过$\text{LS}_+$算子迭代计算稳定集多面体所需的最少迭代次数）的最小图。我们引入一种广义顶点拉伸操作，该操作在生成$\text{LS}_+$-最小图方面具有潜力，并研究其性质。我们还提供了若干新的$\text{LS}_+$-最小图，其中最为显著的是首个已知的具有$\text{LS}_+$-秩$4$的$12$顶点图实例，这标志着自2006年Escalante、Montelar和Nasini发现具有$\text{LS}_+$-秩$3$的$9$顶点图以来，该方向上的首次进展。