of a $k$-tree,

-   minimum integer $k$ such that $G$ is a spanning subgraph of a chordal graph with no $(k+2)$-clique,

-   separation number,

-   branchwidth,

-   tangle number,

-   lexicographic tree product number,

-   cartesian tree product number,

-   linkedness,

-   well-linked number,

-   maximum order of a grid minor,

-   maximum order of a grid-like-minor,

-   Hadwiger number of the Cartesian product $G\square K_2$ (viewed as a function of $G$),

-   fractional Hadwiger number,

-   $r$-integral Hadwiger number for each $r\geq 2$.

@Fox11 states (without proof) a theorem similar to Theorem \[theorem:pttt\] with the parameters treewidth, bramble number, separation number, maximum order of a grid minor, fractional Hadwiger number, and $r$-integral Hadwiger number for each $r\geq 2$. Indeed, this statement of Fox motivated the present paper.

This paper investigates the parameters in Theorem \[theorem:pttt\], showing where these parameters have been useful, and provides proofs that each parameter is tied to treewidth (except in a few cases). In a number of cases we improve known bounds, provide simpler proofs and show that the inequalities presented are tight. The following graph is a key example. Say $n,k$ are integers. Let $\psi_{n,k}$ be the graph with vertex set $A \cup B$, 