Unique and faithful embeddings of projective-planar graphs

A graph G is uniquely embeddable in a surface F2 if for any two embeddings f1,f2: G F2, there exists an isomorphism σ: G G and a homeomorphism h: F2 F2 for which h f1 = f2 σ. A graph G is faithfully embeddable in a surface F2 if G admits an embedding f: G F2 such that for any isomorphism σ: G G, there is a homeomorphism h: F2 F2 with h f = f σ. It will be shown that if a projective-planar graph G is 5-connected and contains a subdivision of the complete graph K6 as its subgraph, then G is uniquely embeddable in a projective plane, and that moreover if G is not isomorphic to K6, then G is faithfully embeddable in a projective plane.