A well-designed interconnection network makes efficient use of scarce communication resources and is used in systems ranging from large supercomputers to small embedded systems on a chip. This paper deals with certain measures of vulnerability in interconnection networks. Let G be a non-complete connected graph and for S ⊆ V (G) let w (G - S) and m (G - S) denote the number of components and the order of the largest component in G - S respectively. The vertex-integrity of G is defined as I(G) = min{ S +m(G-S) : S ⊆ V(G) }. A set S is called an I-set of G if I(G) = S + rn(G - S).The rupture degree of G is defined by r(G) = max{w (G - S) - S - m(G - S) : S ⊆ V(G), w (G - S) ≥ 2}.A set is called an R-set of G if r(G) = w (G - S) - S - m(G - S).In this paper, we compute the rupture degree of complete binary trees, and a class of meshes. We also study the relationship between an I-set and an R-set and find an upper bound for the rupture degree of Hamiltonian graphs.