\(\) We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-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.

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View Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator