CMSO1 Recognizability Characterization for Bounded Cliquewidth

Establish whether recognizability by tree automata is equivalent to mathsf{CMSO}_1-definability for graph properties restricted to graphs of bounded cliquewidth.

Background

The paper uses Courcelles theorem in the forward direction: every mathsf{CMSO}_1-definable property of graphs of bounded cliquewidth is recognized by a multiplicity tree automaton. For bounded-treewidth graphs, the converse equivalence between logical definability and recognizability is known, but the analogous converse for bounded cliquewidth is identified as unresolved. Establishing it would characterize exactly which bounded-cliquewidth graph classes can be recognized by the automata framework used for homomorphism indistinguishability algorithms.

References

Note that for properties of bounded-treewidth graphs \mathsf{CMSO}_2-definability and recognizability by tree automata is equivalent, as conjectured by \textcite{courcelle_monadic_1990} and proven by \textcite{bojanczyk_definability_2016}. For properties of bounded-cliquewidth graphs, only the forward implication is known (\cref{thm:courcelle}); the backward direction is open .

— A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability  (2608.13382 - Curticapean et al., 13 Aug 2026) in Paragraph immediately following Theorem 3.1, Section 5.1 (Bounded cliquewidth)