Forbidden induced subgraphs for toughness

Let F be a family of connected graphs. A graph G is said to be F-free if G is H-free for every graph H in F. We study the relation between forbidden subgraphs in a connected graph G and the resulting toughness of G. In particular, we consider the problem of characterizing the graph families F such that every large enough connected F-free graph is t-tough. In this article, we solve this problem for every real positive number t.