Graphs of Linear Growth have Bounded Treewidth

TL;DR

This paper proves that any class of graphs with linear growth, where small-radius subgraphs are linearly bounded in size, necessarily has bounded treewidth, impacting graph structure theory.

Contribution

It establishes a fundamental link between linear growth and bounded treewidth in graph classes, a previously unknown structural property.

Findings

Graphs with linear growth have bounded treewidth.

Linear growth constrains the complexity of graph structure.

Bounded treewidth facilitates algorithmic applications.

Abstract

A graph class $\mathcal{G}$ has linear growth if, for each graph $G \in \mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with linear growth has bounded treewidth.