MSO Undecidability for Hereditary Classes of Unbounded Clique Width

Seese’s conjecture for finite graphs states that monadic second-order logic (MSO) is undecidable on all graph classes of unbounded clique-width. We show that to establish this it would suffice to show that grids of unbounded size can be interpreted in two families of graph classes: minimal hereditary classes of unbounded clique-width; and antichains of unbounded clique-width under the induced subgraph relation. We explore all the currently known classes of the former category and establish that grids of unbounded size can indeed be interpreted in them. with parameters simply MSO interpretations for ease of readability, and denote them with the uppercase Greek letters Φ , Γ , Ψ , Θ, etc.