Recursive characterization of bounded twin-width

Develop a recursive definition of graph classes of twin-width at most a fixed integer k that fits the inductive framework based on effective and smooth operations used for logic-free extensions of Courcelle’s theorem.

Background

The paper’s main theorem applies to graph classes admitting an effective inductive definition whose operations are smooth with respect to the relevant property. Tree-width, clique-width, and modular width are presented as examples satisfying this framework. The authors explain that twin-width generalizes many established width parameters but do not know a recursive characterization of graphs of twin-width at most k compatible with their method.

A suitable recursive definition would potentially enable the paper’s finite-Hankel-rank approach to yield logic-free Courcelle-type algorithmic results for bounded-twin-width graph classes. The paper notes that existing results establish first-order model checking for bounded twin-width and monadic second-order results for the weaker setting of bounded contraction sequences, leaving the desired extension unresolved.