MSO Undecidability for Hereditary Classes of Unbounded Clique-Width

Abstract

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.

27 pages, 5 figures. The conference version of this paper appeared in Computer Science Logic (CSL) 2022. The only technical addition in this version over the previous version is in Section 4, where the $\omega$-word $\alpha$ is now over the alphabet $\{0,1,2,3\}$ instead of over $\{0,1,2\}$. The other changes are about fixing typos and improving readability