Vulnerability: integrity of a middle graph

A communication network can be considered to be highly vulnerable to disruption if the destruction of a few elements can result in no member's being able to communicate with very many others. This idea suggests the concept of the integrity of a graph. The vertex-integrity of a graph G, denoted I(G), is defined by I(G)=min{|S|+m(G-S)} where m(G-S) denotes the maximum order of a component of G-S. In this paper we consider the integrity of middle graphs of specific families of graphs and combinations of these graphs.