First-order interpretations of bounded expansion classes

TL;DR

This paper introduces structurally bounded expansion graph classes, extending the concept of bounded expansion to dense graphs through first-order interpretations, and characterizes them using shrubdepth as an analogue to treedepth.

Contribution

It generalizes bounded expansion classes to dense graphs via first-order interpretations and provides a characterization using shrubdepth, bridging sparse and dense graph classes.

Findings

Characterization of structurally bounded expansion classes via shrubdepth.

Extension of fixed-parameter tractability results to these classes.

Framework for algorithmic analysis of dense graph classes.

Abstract

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.