Authors’ abstract: “A graph G is said to be m-neighbour-connected if the neighbour-connectivity of the graph, \(K(G)=m\). A graph G is said to be critically m-neighbour-connected if it is m-neighbour-connected and the removal of the closed neighbourhood of any one vertex yields an (m-1)- neighbour-connected subgraph. In this paper, we give some upper bounds of the minimum size of the critically m-neighbour-connected graphs of any fixed order \(\nu\), and show that the number of edges in a minimum critically m-neighbour-connected graph with order \(\nu\) (a multiple of m) is \(\lceil m\nu \rceil\).”