On classes of bounded tree rank, their interpretations, and efficient sparsification
Jakub Gajarský, Rose McCarty
TL;DR
The paper advances the theory of sparse graph classes by linking vertex-centric decompositions ($(r,m)$-rankings) to structural notions like bounded tree rank and bounded expansion, enabling elementary FPT model checking for new graph classes. It provides an explicit, computable decomposition that can be leveraged to understand interpretability and to develop efficient sparsification procedures. A central highlight is the characterization of graph classes interpretable in tree rank $2$ as perturbations of locally almost near-covered graphs, together with a polynomial-time sparsification algorithm that reverses interpretations into a fixed sparse class. Beyond tree rank $2$, the work outlines a pathway to extend these ideas to bounded expansion and interpretable classes, revealing novel tools such as the Rose lemma for $k$-near-twins and a framework for bounded-range interpretations. The results collectively offer new algorithmic levers for FO-model checking and structural understanding of sparse-to-sparse graph transformations.
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 Gajarský et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree.