equal to the size of the largest bag in a minimum width tree decomposition.

Brambles are useful for finding a lower bound on the treewidth of a graph. Consider the following: given a valid tree decomposition $T$ for a graph $G$, then ${\textsf{\textup{tw}}}(G)$ is at most the width of $T$. Brambles provide the equivalent functionality for the lower bound — given a valid bramble of a graph $G$, it follows that the bramble number is at least the order of that bramble, giving us a lower bound on the treewidth. (For examples of this, see @bodlaenderbramble, @lucenabramble and Lemma \[lemma:gridbramble\].)

$k$-Trees and Chordal Graphs
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In certain applications, such as graph drawing [@DMW05; @DiGiacomo] or graph colouring [@KP-DM08; @Albertson-EJC04], it often suffices to consider only the edge-maximal graphs of a given family to obtain a result. The language of $k$-trees and chordal graphs provides an elegant description of the edge-maximal graphs with treewidth at most $k$.

A vertex $v$ in a graph $G$ is *$k$-simplicial* if it has degree $k$ and its neighbours induce a clique. A graph $G$ is a *$k$-tree* if either:

-   $G = K_{k+1}$

-   $G$ has a $k$-simplicial vertex $v$ and $G-v$ is also a 