Every gragh below will refer to a finite simple graph.
There is a necessary and sufficient condition for a nonempty gragh being a line gragh:
A nonempty gragh is a line gragh if and only if its edge set can be partitioned into a set of cliques with the property that any vertex lies in at most two cliques.
And we can find an obvious necessary condition for a nonempty gragh being a line gragh:
If a nonempty gragh $G$ is a line graph,then for any vertex $v$ of $G$,the neighbours of $v$ can be partitioned into at most two vertex sets such that the subgragh of $G$ induced by each vertex set is a clique.
I want to ask if this necessary condition is sufficient for $G$ being a line gragh.If not,please give a counterexample.