MSO Undecidability for Hereditary Classes of Unbounded Clique-Width

TL;DR

This paper investigates the undecidability of monadic second-order logic on hereditary graph classes with unbounded clique-width, linking it to the interpretability of large grids within these classes.

Contribution

It demonstrates that proving MSO undecidability reduces to interpreting unbounded grids in certain hereditary classes, and confirms this interpretability for all known such classes.

Findings

Grids of unbounded size can be interpreted in minimal hereditary classes of unbounded clique-width.

MSO undecidability on these classes is connected to the interpretability of large grids.

The paper establishes interpretability results for all currently known classes of the first category.

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.