RobertsonSymour1991

https://www.doi.org/10.1016/0095-8956(91)90061-N

@article{RobertsonSymour1991,
author = {Neil Robertson and Paul D. Seymour},
doi = {10.1016/0095-8956(91)90061-N},
issue = {2},
journaltitle = {Journal of Combinatorial Theory, Series B},
month = {July},
pages = {153--190},
title = {Graph minors. X. Obstructions to tree-decomposition},
volume = {52},
year = {1991},
}

• page 12 : branch width – A \emph{branch-width} of a hypergraph $G$ is a pair $(T,\tau)$, where $T$ is a ternary tree and $\tau$ is a bijection from the set of leaves of $T$ to $E(G)$. The \emph{order} of an edge $e$ of $T$ is the number of vertices $v$ of $G$ such that there are leaves $t_1,t_2$ of $T$ in different components of $T \setminus e$, with $\tau(t_1),\tau(t_2)$ both incident with $v$. The \emph{width} of $(T,\tau)$ is the maximum order of the edges of $T$, and the \emph{branch-width} $\beta(G)$ of $G$ is the minimum width of all branch-decompositions of $G$ (or 0 if $|E(G)| \le 1$, when $G$ has no branch-decompositions).
• page 16 : treewidth $k$ upper bounds branch width by $\mathcal O(k)$ – (5.1) For any hypergraph $G$, $\max(\beta(G), \gamma(G)) \le \omega(G) + 1 \le \max(\lfloor(3/2)\beta(G)\rfloor, \gamma(G), 1)$.
• page 16 : branch width $k$ upper bounds treewidth by $\mathcal O(k)$ – (5.1) For any hypergraph $G$, $\max(\beta(G), \gamma(G)) \le \omega(G) + 1 \le \max(\lfloor(3/2)\beta(G)\rfloor, \gamma(G), 1)$.