a hitting set for $\beta$ then ${\textsf{\textup{bn}}}(G) \leq |X| \leq sep_c^*(G) < (1-c){\textsf{\textup{bn}}}(G)$, which is a contradiction. So $X$ is not a hitting set for $\beta$. Thus some bramble element of $\beta$ is entirely within a component of $G-X$. Only one such component can contain bramble elements. Call this component $C$. Then we can hit every bramble element of $\beta$ with the vertices of $X$ or the vertices of $H$ inside $C$, that is, $X \cup (H \cap V(C))$ is a hitting set. Since $X$ is a $({\textsf{\textup{sep}}}_c^*(G),H,c)^*$-separator, $|H \cap V(C)| \leq c|H|$. Thus $|X \cup (H \cap V(C))| = |X| + |H \cap V(C)| \leq |X| + c|H| \leq {\textsf{\textup{sep}}}_c^*(G) + c|H| < (1-c)|H| + c|H| = |H|$. Thus $X \cup (H \cap V(C))$ is a hitting set smaller than the minimum hitting set, a contradiction.

Hence, from the above it follows that for $c \in [\frac{1}{2},1)$, $${\textsf{\textup{sep}}}_c^*(G) \leq {\textsf{\textup{sep}}}_c(G) \leq {\textsf{\textup{tw}}}(G) + 1 = {\textsf{\textup{bn}}}(G) \leq \frac{1}{1-c}{\textsf{\textup{sep}}}_c^*(G) \leq \frac{1}{1-c}{\textsf{\textup{sep}}}_c(G).$$

Each of the above inequalities is tight. For a fixed $c \in [\frac{1}{2},1)$, let $k,n$ be integers such that $k > \frac{c}{1-c}+1$ and $n \geq \frac{k-1}{1-c}$. Then ${\textsf{\textup{sep}}}_c^*(\psi_{n,k}) = {\textsf{\textup{sep}}}_c(\psi_{n,k}) = n$, which proves that the first and last 