An equation involving the neighborhood (two-step) and line graphs (Q2760986)

The neighborhood graph \(N(G)\) of a graph \(G\) (also called the two-step graph of \(G\)) is the intersection graph of open neighborhoods of vertices of \(G\) (or, equivalently, \(V(N(G))=V(G)\) and \(xy\in E(N(G))\) if and only if \(x,y\) have a common neighbor in \(G\)). Denote by \(L(G)\) the line graph of \(G\). It is proved that, for a graph \(G\), \(N[L(G)]\) is isomorphic to \(L[N(G)]\) if and only if every component of \(G\) is isomorphic to \(K_1\), \(K_{1,3}\) or \(C_n\) for \(n\geq 3\) and \(n\neq 4\).