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

Authors:
Bruno Courcelle;Sang-il Oum
Affiliations:
Bordeaux 1 University, LaBRI, CNRS, 351 cours de la Libération, F-33405 Talence, France;Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08540, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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Abstract

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.