On Balanced Separators, Treewidth, and Cycle Rank

TL;DR

This paper explores the relationships between graph width parameters, providing refined bounds for treewidth and cycle rank based on the balanced separator number, and demonstrates these bounds are optimal.

Contribution

It establishes tighter bounds linking balanced separator number, treewidth, and cycle rank, improving upon previous results and proving their optimality.

Findings

Treewidth is at least the balanced separator number.

Cycle rank is at most k(1 + log(n/k)).

Bounds are proven to be tight.

Abstract

We investigate relations between different width parameters of graphs, in particular balanced separator number, treewidth, and cycle rank. Our main result states that a graph with balanced separator number k has treewidth at least k but cycle rank at most k(1 + log (n/k)), thus refining the previously known bounds, as stated by Robertson and Seymour (1986) and by Bodlaender et al. (1995). Furthermore, we show that the improved bounds are best possible.