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 inter- preted in two families of graph classes: minimal hereditary classes of unbounded clique-width; and antichains of unbounded clique-width under the induced sub- graph 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.