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). In particular, we provide families of graphs whose $\text{LS}_+$-rank is asymptotically a linear function of its number of vertices, which is the least 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胞体关于Lovász–Schrijver SDP算子$\text{LS}_+$的提升投影秩，特别关注寻找具有高$\text{LS}_+$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胞体的最少次数）的相对小规模图。具体而言，我们提供了一些图族，其$\text{LS}_+$秩渐近地与其顶点数呈线性关系，这是除常数因子改进外可能达到的最小增长（此前该方向的最佳结果来自1999年，仅得到$\text{LS}_+$秩随顶点数平方根增长的图族）。我们还提供了多个新的$\text{LS}_+$极小图，最值得注意的是一个具有12个顶点且$\text{LS}_+$秩为4的图，并研究了顶点拉伸操作的性质，该操作在生成$\text{LS}_+$极小图方面显示出潜力。