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.

翻译：Seese关于有限图的猜想指出，在所有无界团宽度的图类上，一元二阶逻辑（MSO）是不可判定的。我们证明，要建立这一结论，只需证明无界大小的网格可在两类图类中解释：无界团宽度的极小遗传图类，以及在导出子图关系下构成反链的无界团宽度图类。本文探索了目前已知的所有属于前一类别的图类，并证实无界大小的网格确实可在其中解释。