Graph transformations definable in logic can be described using the notion of transductions. By understanding transductions as a basic embedding mechanism, which captures the possibility of encoding one graph in another graph by means of logical formulas, we obtain a new perspective on the landscape of graph classes and of their properties. The aim of this survey is to give a comprehensive presentation of this angle on structural graph theory. We first give a logic-focused overview of classic graph-theoretic concepts, such as treedepth, shrubdepth, treewidth, cliquewidth, twin-width, bounded expansion, and nowhere denseness. Then, we present recent developments related to notions defined purely through transductions, such as monadic stability, monadic dependence, and classes of structurally sparse graphs.
@article{arxiv.2501.04166,
title = {Graph classes through the lens of logic},
author = {Michał Pilipczuk},
journal= {arXiv preprint arXiv:2501.04166},
year = {2025}
}