On the complexity of recognizing tough graphs

Abstract We consider the relationship between the minimum degree δ of a graph and the complexity of recognizing if a graph is t -tough. Let t ⩾1 be a rational number. We first show that if δ ( G )⩾ tn /( t +1), then G is t -tough. On the other hand, for any fixed e>0, we show that it is NP-hard to determine if G is t -tough, even for the class of graphs with δ ( G )⩾( t /( t +1)− e ) n . In particular, for any fixed c G with δ ( G )⩾ cn .