Minimum Tenacity of Toroidal graphs

The tenacity of a graph G, T(G), is dened by T(G) = minfjSj+(G-S) !(G-S) g, where the minimum is taken over all vertex cutsets S of G. We dene (G-S) to be the number of the vertices in the largest component of the graph G􀀀S, and !(G-S) be the number of components of G 􀀀 S.In this paper a lower bound for the tenacity T(G) of a graph with genus (G) is obtained using the graph's connectivity (G). Then we show that such a bound for almost all toroidal graphs is best possible.