As part of a more general categorization of the class of line graphs of linear 3-uniform hypergraphs with minimum vertex degree at least 19, Metelsky and Tyshkevich (1997) established that a graph with minimum vertex degree at least 5 is a line graph iff it does not contain any of a subset of 6 of the Beineke graphs as an induced subgraph.
These graphs, illustrated above, are known in this work as Metelsky graphs and are implemented in the Wolfram Language as GraphData["Metelsky"].
