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

Bruno Courcelle, Sang-il Oum
2007 Journal of combinatorial theory. Series B (Print)  
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
more » ... most k for any fixed k. The proofs use isotropic systems.
doi:10.1016/j.jctb.2006.04.003 fatcat:3dfb5tqexndablnknwykkbcdlu