Recognizing tough graphs is NP-hard

We consider only undirected graphs without loops or multiple edges. Our terminology and notation will be standard except as indicated; a good reference for any undefined terms is [2]. We will use c(G) to denote the number of components of a graph G. Chvtital introduced the notion of tough graphs in [3]. Let t be any positive real number. A graph G is said to be t-tough if tc(G-X)5 JXJ for all Xz V(G) with c(G-X)> 1. The interest in t-tough graphs stems primarily from their connection with the existence of Hamiltonian cycles. It is easy to see that a necessary condition for G to be Hamiltonian is that G is l-tough. In fact, the hypothesis that G is l-tough is often employed in theorems giving sufficient conditions for a graph to be Hamiltonian (e.g., [9,1]). On the other hand, Chvatal [3] conjectured that there exists a positive constant to such that every to-tough graph is Hamiltonian. While this conjecture remains open, it is now known [6] that to cannot be smaller than 2. Moreover, recent results [6] and [l, Corollary 161 suggest the possibility that every 2-tough graph is Hamiltonian.