Abstract
LetS be a compact surface with possibly non-empty boundary ϖS and letG be a graph. LetK be a subgraph ofG embedded inS such that ϖS⊑K. Anembedding extension ofK toG is an embedding ofG inS which coincides onK with the given embedding ofK. Minimal obstructions for the existence of embedding extensions are classified in cases whenS is the projective plane or the Mobius band (for several “canonical” choices ofK). Linear time algorithms are presented that either find an embedding extension, or return a “nice” obstruction for the existence of extensions.
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