Abstract: For P a simplicial polytope of dimension d with v vertices and e edges, the quantity g_2(P) := e - dv + {d+1 choose 2} is known to be non-negative for d>2 and equals zero for d=3. Assume d>3. Kalai proved that the simplicial polytopes with g_2 = 0 are the stacked polytopes; these are the extremal examples in Barnette's lower bound theorem. We characterize the g_2 = 1 case. The proof uses topological tools and graph rigidity theory, which I'll explain. This is joint work with Eyal Novinsky.