An isolated toughness condition for graphs to be fractional (k, m)-deleted graphs

Let i(G) be the number of isolated vertices in a graph G. As an important parameter in computer networks which measures the vulnerability of network, isolated toughness of graph G is defines as I(G) = min{vertical bar S vertical bar/i(G - S)vertical bar S subset of V(G), i(G - S) >= 2} if G is not complete; otherwise, I(G) = infinity. In this paper, we study the relationship between isolated toughness and fractional (k, m)-deleted graphs and determine that G is a fractional (k, m)-deleted graph if I (G) > k + m/k. At last, we propose an open problem for the future studying.