Tenacious Graph is NP-hard

The tenacity of a graph G, T(G), is defined by T(G)=min{∣S∣+τ(G−S)ω(G−S)}, where the minimum is taken over all vertex cutsets S of G. We define τ(G−S) to be the number of the vertices in the largest component of the graph G−S, and ω(G−S) be the number of components of G−S. In this paper we consider the relationship between the minimum degree δ(G) of a graph and the complexity of recognizing if a graph is T-tenacious. Let T≥1 be a rational number. We first show that if δ(G)≥TnT+1, then G is T-tenacious. On the other hand, for any fixed ϵ>0, we show that it is NP-hard to determine if G is T-tenacious, even for the class of graphs with δ(G)≥(TT+1−ϵ)n.