Exactness of Parrilo's conic approximations for copositive matrices and associated low order bounds for the stability number of a graph

Abstract: De Klerk and Pasechnik (2002) introduced the bounds $\vartheta^{(r)}(G)$ ($r\in \mathbb{N}$) for the stability number $\alpha(G)$ of a graph $G$ and conjectured exactness at order $\alpha(G)-1$: $\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)$. These bounds rely on the conic approximations $\mathcal{K}n^{(r)}$ by Parrilo (2000) for the copositive cone $\text{COP}_n$. A difficulty in the convergence analysis of $\vartheta^{(r)}$ is the bad behaviour of the cones $\mathcal{K}_n^{(r)}$ under adding a zero row/column: when applied to a matrix not in $\mathcal{K}^{(0)}_n$ this gives a matrix not in any ${\mathcal{K}}^{(r)}{n+1}$, thereby showing strict inclusion $\bigcup_{r\ge 0}{\mathcal{K}}^{(r)}_n\subset \text{COP}_n$ for $n\ge 6$. We investigate the graphs with $\vartheta^{(r)}(G)=\alpha(G)$ for $r=0,1$: we algorithmically reduce testing exactness of $\vartheta^{(0)}$ to acritical graphs, we characterize critical graphs with $\vartheta^{(0)}$ exact, and we exhibit graphs for which exactness of $\vartheta^{(1)}$ is not preserved under adding an isolated node. This disproves a conjecture by Gvozdenovi\'c and Laurent (2007) which, if true, would have implied the above conjecture by de Klerk and Pasechnik.