On minimally tough chordal graphs

Katona and Varga showed that for any rational number $t \in (1/2,1]$, no chordal graph is minimally $t$-tough, while Katona and Khan characterized all minimally $t$-tough, chordal graphs with $t \le 1/2$. We conjecture that no chordal graph is minimally $t$-tough for any $t>1$ and prove several results supporting the conjecture. In particular, we show that for any $t>1/2$, no strongly chordal graph is minimally $t$-tough%, no split graph is minimally $t$-tough, and no chordal graph with a universal vertex is minimally $t$-tough.