Abstrakt
A partially embedded graph (or PEG) is a triple (G, H, H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a PEG (G, H, H) is planar if the graph G has a planar embedding that extends the embedding H.
We introduce a containment relation of PEGS analogous to graph minor containment, and characterize the minimal non-planar PEGS with respect to this relation. We show that all the minimal non-planar PEGS except for finitely many belong to a single easily recognizable and explicitly described infinite family.
We also describe a more complicated containment relation which only has a finite number of minimal non-planar PEGS. Furthermore, by extending an existing planarity test for PEGS, we obtain a polynomial-time algorithm which, for a given PEG, either produces a planar embedding or identifies an obstruction.