The Treewidth of Line Graphs

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.