Exactness of Parrilo’s Conic Approximations for Copositive Matrices and Associated Low Order Bounds for the Stability Number of a Graph

De Klerk and Pasechnik introduced in 2002 semidefinite bounds ϑ ( r ) ( G ) ( r ≥ 0 ) for the stability number α ( G ) of a graph G and conjectured their exactness at order r = α ( G ) − 1 . These bounds rely on the conic approximations K n ( r ) introduced by Parrilo in 2000 for the copositive cone COP n . A difficulty in the convergence analysis of the bounds is the bad behavior of Parrilo’s cones under adding a zero row/column: when applied to a matrix not in K n ( r ) this gives a matrix that does not lie in any of Parrilo’s cones, thereby showing that their union is a strict subset of the copositive cone for any n ≥ 6 . We investigate the graphs for which the bound of order r ≤1 is exact: we algorithmically reduce testing exactness of ϑ ( 0 ) to acritical graphs, we characterize the critical graphs with ϑ ( 0 ) exact, and we exhibit graphs for which exactness of ϑ ( 1 ) is not preserved under adding an isolated node. This disproves a conjecture posed by Gvozdenović and Laurent in 2007, which, if true, would have implied the above conjecture by de Klerk and Pasechnik.