Vertex-minors, monadic second-order logic, and a conjecture by Seese

We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width. We also obtain a polynomial-time algorithm to check that the rank-width of a graph is at most k for any fixed k. The proofs use isotropic systems.