Non-extendible finite polycycles

We give a fairly simple proof of a result announced in [2] that there are only seven non-extendible finite $(r,q)$-polycycles: the tetrahedron without a face, the cube without a face, the octahedron without a face, the dodecahedron without a face, the icosahedron without a face, the split-vertex octahedron and the split-vertex icosahedron.