The Fibonacci cube Γ n is the subgraph of the n-dimensional cube Q n induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for γ t (Γ n), n ≤ 12. Consequently, γ t (Γ n) ≤ 2F n−10 + 21F n−8 holds for n ≥ 11, where F n are the Fibonacci numbers. It is proved that if n ≥ 9, then γ t (Γ n) ≥ (F n+2 − 11)/(n − 3) − 1. Using integer linear programming exact values for the 2packing number, connected domination number, paired domination number, and signed domination number of small Fibonacci cubes and hypercubes are obtained. A conjecture on the total domination number of hypercubes asserting that γ t (Q n) = 2 n−2 holds for n ≥ 6 is also disproved in several ways.