The main purpose of this work is to characterize the graphs whose closed neighbourhood matrices are perfect since this property implies the resolution of some packing problems in polynomial time. Given a 0-1 matrix M, Q(M) denotes the graph whose cliques have their incidence vectors as the rows of M. A 0-1 matrix M is perfect if it is the clique-node matrix of Q(M) and Q(M) is a perfect graph. First, we define a set T of seven graphs and prove that N[G] is a clique-node matrix if and only if every node induced subgraph of G in the set T, has a common neighbour in G. Second, we define two families of graphs, called H-graphs and A-graphs, and prove that Q(N[G]) does not have an induced odd hole if and only if G has no H-subgraph for some additional conditions on G. Finally, we show that Q(N[G]) does not have an induced odd antihole if and only if G has neither an A-subgraph nor a web graph Wt4t+3 in suitable sets of nodes of V(G), thus completing a characterization of graphs with perfect closed neighbourhood matrix.