sufficient to prove that $\Delta_{W}$ (equivalently $(\bar{G})_W$) is connected for any $W\subset [n]$ with $|W|=t+2$.

Since $\index(I)=t$, [@EGHP Theorem 2.1] implies that there exists a minimal cycle $C$ of length $t+3$ in $\bar{G}$. Without loss of generality we may suppose that $V(C)=\{1, 2, \ldots, t+3\}$ and $E(C)=\{\{i, i+1\} \:\ 1\leq i\leq t+2\}\cup \{\{1, t+3\}\}$.

Let $W$ be a subset of $[n]$ with $|W|=t+2$. We consider different cases for $W$ and prove in each case that $(\bar{G})_W$ is connected.

First assume that $W\subset V(C)$. Since $V(C)$ has just one vertex more than $W$ we see that $(\bar{G})_W$ is a line graph and thus is connected.

It remains to consider the case $W\setminus V(C)\neq \emptyset$.

We first claim that for all $j\in W\setminus V(C)$ and all $i\in V(C)$ we have $\{j, i\}\in E(\bar{G})$.

Indeed, suppose that $\{j, i\}\notin E(\bar{G})$ for some $j\in W\setminus V(C)$ and some $i\in V(C)$. Let $W'=V(C)\cup \{j\}$ and consider $\Delta_{W'}$. Note that $\Delta_{W'}$, as a topological space, is homotopy equivalent either to $\mathcal{S}_1$ or to $\mathcal{S}_1$ together with an isolated point. The second case happens only if $\{j, i\}\notin E(\bar{G})$ for all $i\in V(C)$. In either case we see that $\widetilde{H}_{1}(\Delta_{W'}; K)\neq 0$. Now Hochster’s formula implies that 