considering components in $G-X$, separators are defined as partitioning the vertex set of $G-X$ into exactly two parts $A$ and $B$, such that no edge has an endpoint in both parts and $|A \cap S|,|B \cap S| \leq c|S|$. (In fact, @Bodlaender-TCS98 uses both this definition and the standard “components of $G-X$" definition as the difference between type-1 and type-2 separators.) As long as $c \geq \frac{2}{3}$, this is equivalent to considering the components, since Lemma \[lemma:comptrick\] and Corollary \[corollary:comptrick\] allow partitioning of the components into parts $A$ and $B$. However, for lower values of $c$ this no longer holds, for example, if $c=\frac{1}{2}$, it is possible that each component contains exactly $\frac{1}{3}$ of the vertices of $S$, so there is no acceptable partition into $A$ and $B$. As a result, $c=\frac{2}{3}$ and $c=\frac{1}{2}$ are the most “natural" choices for $c$.

Fortunately, ${\textsf{\textup{sep}}}_c(G),{\textsf{\textup{sep}}}_c^*(G),{\textsf{\textup{sep}}}_{c'}(G)$ and ${\textsf{\textup{sep}}}_{c'}^*(G)$ are all tied for all $c,c' \in [\frac{1}{2},1)$. [^4]

@RS-GraphMinorsII-JAlg86 proved that $${\textsf{\textup{sep}}}_{\frac{1}{2}}(G)\leq {\textsf{\textup{tw}}}(G)+1\enspace.$$ (Of course, they did not use our notation.) @RS-GraphMinorsII-JAlg86 [@RS-GraphMinorsXIII-JCTB95] also proved that $${\textsf{\textup{tw}}}(G)+1\leq 4\,{\textsf{\textup{sep}}}_{\frac{2}{3}}(G) - 2\enspace.$$ (@Reed97 [@Reedktreefinder] gives a more accessible proof of this upper bound.) We now provide a series of lemmas to prove a slightly stronger result, 