inequalities are tight. (See @thesis for a proof of this result.) The remaining two inequalities are tight due to the complete graph $K_n$.

Branchwidth and Tangles
=======================

A *branch decomposition* of a graph $G$ is a pair $(T, \theta)$ where $T$ is a tree with each node having degree 3 or 1, and $\theta$ is a bijective mapping from the edges of $G$ to the leaves of $T$. A vertex $x$ of $G$ is *across* an edge $e$ of $T$ if there are edges $xy$ and $xz$ of $G$ mapped to leaves in different subtrees of $T-e$. The *order* of an edge $e$ of $T$ is the number of edges of $G$ across $e$. The *width* of a branch decomposition is the maximum order of an edge. Finally, the *branchwidth* ${\textsf{\textup{bw}}}(G)$ of a graph $G$ is the minimum width over all branch decompositions of $G$. Note that if $|E(G)| \leq 1$, there are no branch decompositions of $G$, in which case we define ${\textsf{\textup{bw}}}(G)=0$. @RS-GraphMinorsX-JCTB91 first defined branchwidth, where it was defined more generally for hypergraphs; here we just consider the case of simple graphs. Tangles were first defined by @RS-GraphMinorsX-JCTB91. Their definition is in terms of sets of separations 