Efficient reversal of transductions of sparse graph classes
This work investigates the efficient approximate inversion of first-order definable graph transductions where the source graphs belong to sparse classes. For graph classes with structurally bounded expansion, we design an O(n⁴)-time algorithm that, given an input graph G, reconstructs a vertex-colored graph H of bounded expansion such that G can be recovered from H via a first-order interpretation. Our result resolves an open problem posed by Gajarský et al., significantly weakening the required conditions for invertible transductions to merely unary stability and inherent linear neighborhood complexity. As a consequence, we obtain a novel equivalent characterization of structurally bounded expansion classes, thereby establishing a deep connection between structural graph properties and logical definability.