Geodesic
Non-extendible finite polycycles
Izvestiya. Mathematics , Tome 70 (2006) no. 1, pp. 1-18

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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. 
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M. Deza; S. V. Shpektorov; M. I. Shtogrin. Non-extendible finite polycycles. Izvestiya. Mathematics , Tome 70 (2006) no. 1, pp. 1-18. http://geodesic.mathdoc.fr/item/IM2_2006_70_1_a0/