Abstract. By L(G) we denote a line graph of G. We set L0(G)=G. Then for i>0 the i-iterated line graph of G is Li(G)=L(Li-1(G)). Further, by C(G) we denote the subgraph of G induced by central vertices.
Let G be a graph such that L2(G) is not empty. Then there is a supergraph H of G such that

C(Li(H))=Li(G) for i=0,1,2

and
C(L3(H))=L3(G) if L3(G) is not empty and G is triangle-free.

This result is, in a sense, best possible since we provide an infinite class of graphs G such that C(L3(H)) is not L3(G) for any supergraph H of G.