  It is shown that up to dimension four, the toric ideal of a quiver polytope
is generated in degree two, with the only exception of the four-dimensional
Birkhoff polytope. As a consequence, B{\o}gvad's conjecture holds for quiver
polytopes of dimension at most four. In arbitrary dimension, the toric ideal of
a compressed polytope is generated in degree two if the polytope has no
neighbouring singular vertices. Furthermore, the toric ideal of a compressed
polytope with at most one singular vertex has a quadratic Gr\"obner basis.
