Minimally tough chordal graphs with toughness at most $1\mathrm{/}2$

Let $t$ be a positive real number. A graph is called \emph{$t$-tough} if the removal of any vertex set $S$ that disconnects the graph leaves at most $\mathrm{\mid }S\mathrm{\mid }\mathrm{/}t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. A graph is \emph{chordal} if it does not contain an induced cycle of length at least $4$. We characterize the minimally $t$-tough, chordal graphs for all $t\le 1\mathrm{/}2$. As a corollary, a characterization of minimally $t$-tough, interval graphs is obtained for $t\le 1\mathrm{/}2$.