On Rank-Monotone Graph Operations and Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy

Abstract: 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 finding and characterizing the smallest graphs with a given $\text{LS}+$-rank (the needed number of iterations of the $\text{LS}+$ operator on the fractional stable set polytope to compute the stable set polytope). We introduce a generalized vertex-stretching operation that appears to be promising in generating $\text{LS}+$-minimal graphs and study its properties. We also provide several new $\text{LS}+$-minimal graphs, most notably the first known instances of $12$-vertex graphs with $\text{LS}+$-rank $4$, which provides the first advance in this direction since Escalante, Montelar, and Nasini's discovery of a $9$-vertex graph with $\text{LS}_+$-rank $3$ in 2006.