MSO Undecidability for some Hereditary Classes of Unbounded Clique-Width
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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 a number of known examples of the former category and establish that grids of unbounded size can indeed be interpreted in them.
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