Let t be a positive real number. A graph is called t-tough, if the
removal of any cutset S leaves at most ∣S∣/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. In this paper we investigate the
minimum degree and the recognizability of minimally t-tough graphs in the
class of chordal graphs, split graphs, claw-free graphs and 2K2​-free graphs