Efficiently Computable Preimages for First-Order Transductions

Construct, for every first-order transduction $\tau$ of the class of planar graphs or of a class admitting product structure, a first-order transduction $\tau'$ subsuming $\tau$ such that every graph in $\tau(\mathcal C)$ has an efficiently computable preimage in $\mathcal C$ whose image under $\tau'$ contains that graph.

Background

The paper observes that its constructive proofs can compute product-structure expressions when the source graph and the coloring used by the transduction are supplied. However, recovering an appropriate source graph from an output graph is a separate algorithmic obstacle in first-order model checking. The final question asks whether the transduction can be replaced by a subsuming one that guarantees efficiently computable source preimages.

References

Let $\ca C$ be the class of planar graphs (or a class admitting a product structure), and $\tau$ be a first-order transduction. Can we find a first-order transduction $\tau'$ (of $\ca C$) subsuming $\tau$ such that, for every graph $G\in\tau(\ca C)$ there is an efficiently computable preimage $G'\in\ca C$ such that $G \in \tau'(G')$?

— Transductions of Graph Classes Admitting Product Structure  (2501.18326 - Hliněný et al., 30 Jan 2025) in Concluding Remarks, Question following \ref{que:expression-sparse}