The Treewidth of Line Graphs

TL;DR

This paper investigates the treewidth of line graphs, establishing its relation to vertex congestion in tree embeddings, and provides exact and improved bounds for various graph classes.

Contribution

It introduces a novel equivalence between line graph treewidth and vertex congestion, enabling precise bounds and exact calculations for specific graph families.

Findings

Exact treewidth for line graphs of complete graphs

Sharp lower bounds based on degree measures

Improved upper bounds on line graph treewidth

Abstract

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to determining the minimum vertex congestion of an embedding of $G$ into a tree. Using this result, we prove sharp lower bounds in terms of both the minimum degree and average degree of $G$. These results are precise enough to exactly determine the treewidth of the line graph of a complete graph and other interesting examples. We also improve the best known upper bound on the treewidth of a line graph. Analogous results are proved for pathwidth.