The vulnerability of the communication network measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. Cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points can cause interrupt service for long periods of time. High levels of service dependability have traditionally characterised communication services. In communication networks, requiring greater degrees of stability or less vulnerability. If we think of graph G as modelling a network, the neighbour-integrity and edge-neighbour-integrity of a graph, which are considered as the neighbour vulnerability, are two measures of graph vulnerability. In the neighbour-integrity, it is considered that any failure vertex affects its neighbour vertices. In the edge-neighbour-integrity it is consider that any failure edge affects its neighbour edges.
In this paper we study classes of recursive graphs that are used to design communication networks and represent the molecular structure, and we show neighbour-integrity (vertex and edge) among the recursive graphs.