The monadic second-order logic of graphs XIII: Graph drawings with edge crossings

We introduce finite relational structures called sketches, that represent edge crossings in drawings of finite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges that may cross one another and that may cross the edges of the frame. We prove the 3-Edge Theorem stating that a structure of appropriate type with a frame is a sketch if and only if every induced substructure representing the frame and at most 3 edges not in the frame is a sketch.