Projective Plane and Möbius Band Obstructions

Let S be a compact surface with possibly non-empty boundary ∂S and let G be a graph. Let K be a subgraph of G embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embedding extensions are classified in cases when S is the projective plane or the Möbius band (for several “canonical” choices of K). Linear time algorithms are presented that either find an embedding extension, or return a “nice” obstruction for the existence of extensions.