  The Lov\'asz theta function $\theta(G)$ provides a very good upper bound on
the stability number of a graph $G$. It can be computed in polynomial time by
solving a semidefinite program (SDP), which also turns out to be fairly
tractable in practice. Consequently, $\theta(G)$ achieves a hard-to-beat
trade-off between computational effort and strength of the bound. Indeed,
several attempts to improve the theta bound are documented, mainly based on
playing around the application of the $N_+(\cdot)$ lifting operator of Lov\'asz
and Schrijver to the classical formulation of the maximum stable set problem.
Experience shows that solving such SDP-s often struggles against practical
intractability and requires highly specialized methods. We investigate the
application of such an operator to two different linear formulations based on
clique and nodal inequalities, respectively. Fewer inequalities describe these
two and yet guarantee that the resulting SDP bound is at least as strong as
$\theta(G)$. Our computational experience, including larger graphs than those
previously documented, shows that upper bounds stronger than $\theta(G)$ can be
accessed by a reasonable additional effort using the clique-based formulation
on sparse graphs and the nodal-based one on dense graphs.
