Obstructions for simple embeddings

Suppose that K ⊆ G is a graph embedded in some surface and F is a face of K with singular branches e and f such that F ∪ ∂F is homeomorphic to the torus minus an open disk. An embedding extension of K to G is a simple embedding if each K-bridge embedded in F is attached to at most one appearance of e and at most one appearance of f on ∂F . Combinatorial structure of minimal obstructions for existence of simple embedding extensions is described. Moreover, a linear time algorithm is presented that either finds a simple embedding, or returns an obstruction for existence of such embeddings.