The monadic second-order logic of graphs XII: planar graphs and planar maps

We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding of a graph can be specified by a relational structure called a (combinatorial) map, which is a graph enriched with a circular ordering of the edges incident with each vertex. This order represents a planar embedding of the neighbourhood of each vertex. For each connected map one can define a linear order on its vertices by formulas of monadic second-order logic. Hence, we have for planar graphs, some kind of equivalence between linear orderings and planar embeddings.