Separation Number and Treewidth, Revisited

TL;DR

This work establishes a linear upper bound on the treewidth in terms of the separation number by a constructive proof that avoids brambles, havens, and network-flow machinery. It introduces generalized $W$-balanced separations and uses Menger-type path arguments to iteratively build a tree decomposition, ensuring all bags stay within a constant factor of the separation number. The core technical contribution is a careful analysis (via $W$-sequences and a key bound) that enables a recursive assembly of decompositions on subgraphs around a balanced separation. As a result, the paper provides an explicit constant $c=7915/139<56.943$ with a polynomial-time construction of the decomposition, offering a practical, threshold-based relation between $tw(G)$ and $sn(G)$.

Abstract

We give a constructive proof of the fact that the treewidth of a graph $G$ is bounded by a linear function of the separation number of $G$.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2: robertson.seymour:graph
• Theorem 3: Menger's Theorem
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7