Daniel J. Harvey

Parameters Tied to Treewidth

Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This article surveys several graph parameters tied to treewidth, including separation number, tangle number, well-linked number, and Cartesian tree product number. We review many results in the literature showing these parameters are tied to treewidth. In a number of cases we also improve known bounds, provide simpler proofs, and show that the inequalities presented are tight.

Cycles of given size in a dense graph

We generalize a result of Corradi and Hajnal and show that every graph with average degree at least

A linear-time algorithm for finding a complete graph minor in a dense graph

\frac{4}{3}kr

The treewidth of line graphs

contains

Average degree conditions forcing a minor

k

Treewidth of the Line Graph of a Complete Graph

vertex disjoint cycles, each of order at least

On Treewidth and Graph Minors

r

Treewidth of the Kneser graph and the Erdős-Ko-Rado Theorem

, as long as

Treewidth of Line Graphs

k \geq 6

Treewidth of the Line Graph of Complete and Complete Multipartite Graphs

. This bound is sharp when r=3.