A graph G is minimally t -tough if the toughness of G is t and the deletion of any edge from G decreases the toughness. Kriesell conjectured that for every minimally 1 -tough graph the minimum degree δ ( G ) = 2 . We show that in every minimally 1 -tough graph δ ( G ) ≤ n 3 + 1 . We also prove that every minimally 1 -tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number t any graph can be embedded as an induced subgraph into a minimally t -tough graph.