  Let k>0 be an integer, let H be a minor-minimal graph in the projective plane
such that every homotopically non-trivial closed curve intersects H at least k
times, and let G be the planar double cover of H obtained by lifting G into the
universal covering space of the projective plane, the sphere. We prove that G
is minor-minimal of branch-width 2k. We also exhibit examples of minor-minimal
planar graphs of branch-width 6 that do not arise this way.
