V(G)$ with $|X| \leq k$ such that no component of $G-X$ contains more than $c|S|$ vertices of $S-X$. Define ${\textsf{\textup{sep}}}_c^*(G)$ analogously to ${\textsf{\textup{sep}}}_c(G)$, but with respect to these variant separators. It follows from the definition that ${\textsf{\textup{sep}}}_c^*(G) \leq {\textsf{\textup{sep}}}_c(G)$.

Separators can be seen as a generalisation of the ideas presented in the famous planar separator theorem [@LT79], which essentially states that a planar graph $G$ with $n$ vertices has a $(O(\sqrt{n}),V(G),\frac{2}{3})^*$-separator. Unfortunately, the precise definition of a separator and the separation number is inconsistent across the literature. The above definition is an attempt to unify all existing definitions. @RS-GraphMinorsII-JAlg86 gave the first lower bound on ${\textsf{\textup{tw}}}(G)$ in terms of separators, though they do not use the term, nor do they give an explicit definition of separation number. This definition is equivalent to our standard definition but with $c$ fixed at $\frac{1}{2}$. @GM-JCTB, give the above variant definition, with $c$ fixed at $\frac{1}{2}$, and instead call it a *balanced separator*. @Reed97 defines separators using our standard definition, with $c=\frac{2}{3}$. @Bodlaender-TCS98 defines type-1 and type-2 separators, which have variable proportion (i.e. allow for different values of $c$), but are not defined on sets other than $V(G)$. Sometimes [@Fox11; @GM-JCTB; @Bodlaender-TCS98] instead of 