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 a search for relatively small graphs with high $\text{LS}_+$-rank (the least number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\text{LS}_+$-rank is asymptotically a linear function of its number of vertices, which is the best possible up to improvements in the constant factor (previous best result in this direction, from 1999, yielded graphs whose $\text{LS}_+$-rank only grew with the square root of the number of vertices). We also provide several new $\text{LS}_+$-minimal graphs, most notably a $12$-vertex graph with $\text{LS}_+$-rank $4$, and study the properties of a vertex-stretching operation that appears to be promising in generating $\text{LS}_+$-minimal graphs.

翻译：我们研究图稳定集多面体关于Lovász-Schrijver SDP算子$\text{LS}_+$的提升-投影秩，重点关注寻找具有高$\text{LS}_+$-秩（即在分数稳定集多胞体上迭代$\text{LS}_+$算子以计算稳定集多面体的最少迭代次数）的相对小规模图。我们构造了图族，其$\text{LS}_+$-秩渐近地达到顶点数的线性函数，这是常数因子改进意义下的最优结果（此前该方向的最佳结果来自1999年，所得图的$\text{LS}_+$-秩仅随顶点数的平方根增长）。我们还提供了若干新的$\text{LS}_+$-极小图，最值得注意的是一个具有$\text{LS}_+$-秩为4的12顶点图，并研究了在生成$\text{LS}_+$-极小图方面具有前景的顶点拉伸操作的性质。