  Let $t$ be a positive real number. A graph is called $t$-tough if the removal
of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$
components, and all graphs are considered 0-tough. The toughness of a graph is
the largest $t$ for which the graph is $t$-tough, whereby the toughness of
complete graphs is defined as infinity. 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. In this paper, we investigate the minimum degree and
the recognizability of minimally $t$-tough graphs in the classes of chordal
graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.
