extra connectivity of lexicographic product graphs

Lexicographic product of two graphs g1 = (v (g1); e(g1)) and g2 = (v (g2); e(g2)), denoted byg1 × g2, has vertex set v (g1 × g2) = v (g1) × v (g2) and edge set e(g1 × g2) = f(a; b)(x; y) j ax 2e(g1) or a = x; by 2 e(g2)g. a vertex-cut s of a graph g is called an e-extra vertex-cut if g − sis disconnected and every componrnt of g − s has more than e vertices. the e-extra connectivity of agraph g, κe(g), is defined as the minimum cardinality over all e-extra vertex-cuts of g. in this paper weinvestigate e-extra connectivity of g1 × g2 and give some results.