Finite convergence of sum-of-squares hierarchies for the stability number of a graph

We investigate a hierarchy of semidefinite bounds \vargamma (r)(G) for the stability number \alpha (G) of a graph G, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim., 12 (2002), pp. 875--892], who conjectured convergence to \alpha (G) in r = \alpha (G) 1 steps. Even the weaker conjecture claiming finite convergence is still open. We establish links between this hierarchy and sum-of-squares hierarchies based on the Motzkin--Straus formulation of \alpha (G), which we use to show finite convergence when G is acritical, i.e., when \alpha (G \setminus e) = \alpha (G) for all edges e of G. This relies, in particular, on understanding the structure of the minimizers of the Motzkin--Straus formulation and showing that their number is finite precisely when G is acritical. Moreover we show that these results hold in the general setting of the weighted stable set problem for graphs equipped with positive node weights. In addition, as a byproduct we show that deciding whether a standard quadratic program has finitely many minimizers does not admit a polynomial time algorithm unless P=NP.