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

We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lov{\'a}sz--Schrijver SDP operator $\LS_+$, with a particular focus on a search for relatively small graphs with high $\LS_+$-rank (the least number of iterations of the $\LS_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\LS_+$-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 $\LS_+$-rank only grew with the square root of the number of vertices).