Efficient reversal of transductions of sparse graph classes

TL;DR

This paper introduces an efficient algorithm to approximately reverse first-order transductions on sparse graph classes, enabling recovery of original graphs within bounded expansion classes, thus addressing an open problem in graph theory.

Contribution

It provides a new $O(n^4)$-time method to reverse transductions for classes with structurally bounded expansion, under weaker assumptions like monadic stability and linear neighborhood complexity.

Findings

Algorithm works in $O(n^4)$ time for sparse graphs

Reverses transductions to recover graphs in bounded expansion classes

Addresses an open problem in graph transduction theory

Abstract

(First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarsk\'y et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion.