Lacon-, Shrub- and Parity-Decompositions: Characterizing Transductions of Bounded Expansion Classes

TL;DR

This paper introduces lacon-, shrub-, and parity-decompositions to encode graphs from classes with bounded expansion, enabling the transfer of algorithmic properties from sparse to more general dense graph classes.

Contribution

It presents novel decomposition techniques that encode structurally bounded expansion graphs, facilitating the extension of sparse graph properties to denser classes.

Findings

Decompositions encode graphs from bounded expansion classes.

These techniques enable transferring algorithmic properties to dense graph classes.

The methods help lift properties from sparse to structurally sparse graphs.

Abstract

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts. In this paper, we show that one can encode graphs from a class with structurally bounded expansion via lacon-, shrub- and parity-decompositions from a class with bounded expansion. These decompositions are useful for lifting properties from sparse to structurally sparse graph classes.