We study the lift-and-project relaxations of the stable set polytope of graphs generated by $\text{LS}_+$, the SDP lift-and-project operator devised by Lovász and Schrijver. Our focus is on $\ell$-minimal graphs: graphs on $3\ell$ vertices with $\text{LS}_+$-rank $\ell$, i.e., the smallest graphs realizing rank $\ell$. This manuscript makes two complementary contributions. First, we introduce $\text{LS}_+$ certificate packages, a modular framework for certifying membership in $\text{LS}_+$-relaxations using only integer arithmetic and simple, concise calculations, thereby making numerical lower-bound proofs more transparent, reliable, and easier to verify. Second, we apply this framework to a computational search for extremal graphs. We prove that there are at least 49 non-isomorphic 3-minimal graphs and at least 4,107 non-isomorphic 4-minimal graphs, improving the previously known counts of 14 and 588, respectively. Beyond the increase in counts, the new examples sharpen the emerging structural picture: stretched cliques remain central but are not exhaustive, clique number is informative but not decisive, and some extremal graphs exhibit previously unseen graph minor and edge density behaviour. We also determine the smallest vertex-transitive graphs of $\text{LS}_+$-rank $\ell$ for every $\ell \leq 4$.

翻译：我们研究由Lovász和Schrijver提出的SDP提升-投影算子$\text{LS}_+$生成的图稳定集多面体的提升-投影松弛。重点关注$\ell$-极小图：具有$\text{LS}_+$秩$\ell$的$3\ell$个顶点图，即实现秩$\ell$的最小图。本文做出两个互补贡献。首先，我们引入$\text{LS}_+$证书包，这是一种模块化框架，仅使用整数算术和简单、简洁的计算来认证$\text{LS}_+$松弛中的成员资格，从而使数值下界证明更加透明、可靠且易于验证。其次，我们将该框架应用于极值图的计算搜索。我们证明至少存在49个非同构的3-极小图和至少4,107个非同构的4-极小图，分别改进了此前已知的14个和588个计数。除计数增加外，新例子还丰富了涌现的结构图景：拉伸团仍具核心地位但并非穷尽，团数具有信息性但非决定性，且一些极值图展现出此前未见的小图子式与边密度行为。我们还确定了每个$\ell \leq 4$的最小$\text{LS}_+$秩$\ell$顶点传递图。