Treewidth of graphs with balanced separations

TL;DR

This paper establishes a linear relationship between a graph's treewidth and its separation number, showing that graphs with small balanced separations have bounded treewidth, which has implications for graph algorithms and structure theory.

Contribution

It proves that graphs with subgraphs having small balanced separations also have bounded treewidth, establishing a linear bound between these parameters.

Findings

Treewidth is at most 15 times the separation size for such graphs.

Graphs with small balanced separations have bounded treewidth.

The result provides a linear bound linking separation number and treewidth.

Abstract

We prove that if every subgraph of a graph $G$ has a balanced separation of order at most $a$ then $G$ has treewidth at most $15a$. This establishes a linear dependence between the treewidth and the separation number.