Abstract
LetSbe a closed surface with boundary ∂Sand letGbe a graph. LetK⊆Gbe a subgraph embedded inSsuch that ∂S⊆K. Anembedding extensionofKtoGis an embedding ofGinSthat coincides onKwith the given embedding ofK. Minimal obstructions for the existence of embedding extensions are classified in cases whenSis the disk or the cylinder. Linear time algorithms are presented that either find an embedding extension, or return an obstruction to the existence of extensions. These results are to be used as the corner stones in the design of linear time algorithms for the embeddability of graphs in an arbitrary surface and for solving more general embedding extension problems.
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