a graph in a minor-closed class; see @KM-GC07 for a survey of several versions of the Graph Minor Structure Theorem. The most complex version, and the one used in the proof of the Graph Minor Theorem, describes the structure of graphs in a minor-closed class with unbounded treewidth in terms of *tangles*. Robertson and Seymour combined all these ingredients in their proof of the Graph Minor Theorem. The purpose of this paper is to present a number of known graph parameters that are closely related to treewidth, including those mentioned above such as separation number, linkedness, well-linked number and tangle number.

Formally, a *graph parameter* is a real-valued function $\alpha$ defined on all graphs such that $\alpha(G_1)=\alpha(G_2)$ whenever $G_1$ and $G_2$ are isomorphic. Two graph parameters $\alpha(G)$ and $\beta(G)$ are *tied*[^3] if there exists a function $f$ such that for every graph $G$, $$\alpha(G) \leq f(\beta(G)) \text{ and } \beta(G) \leq f(\alpha(G)).$$ Moreover, say that $\alpha$ and $\beta$ are *polynomially tied* if $f$ is a polynomial.

Drawing on results in the literature, we prove the following:

\[theorem:pttt\] The following graph parameters are polynomially tied:

-   treewidth,

-   bramble number,

-   minimum integer $k$ such that $G$ is a spanning subgraph 