On classes of bounded tree rank, their interpretations, and efficient sparsification

TL;DR

This paper introduces efficient decomposition and interpretation techniques for graph classes of bounded tree rank, enabling sparsification and extending previous results to broader classes of graphs.

Contribution

It provides a new decomposition method for bounded tree rank classes, characterizes interpretability in rank 2 classes, and offers an efficient sparsification procedure.

Findings

Efficient algorithms for decomposing bounded tree rank graph classes.

Characterization of graph classes interpretable in rank 2 classes.

Polynomial-time sparsification of interpretable graphs.

Abstract

Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded treedepth, and they are a special case of graph classes of bounded expansion. We introduce a notion of decomposition for these classes and show that these decompositions can be efficiently computed. Also, a natural extension of our decomposition leads to a new characterization and decomposition for graph classes of bounded expansion (and an efficient algorithm computing this decomposition). We then focus on interpretations of graph classes of bounded tree rank. We give a characterization of graph classes interpretable in graph classes of tree rank $2$. Importantly, our characterization leads to an efficient sparsification procedure: For any graph class $C$ interpretable in a efficiently bounded graph class of tree rank at most $2$, there is a polynomial time algorithm that to any $G \in C$ computes a (sparse) graph $H$ from a fixed graph class of tree rank at most $2$ such that $G = I(H)$ for a fixed interpretation $I$. To the best of our knowledge, this is the first efficient "interpretation reversal" result that generalizes the result of Gajarsk\'y et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree.