TOPICS
Metelsky Graphs
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"].
See also
Beineke Graphs, Forbidden Induced Subgraph, Line Graph, Šoltes GraphsExplore with Wolfram|Alpha
References
House of Graphs. Metelsky Graphs. K4 with One Subdivided Edge, Beineke Non-Line Graph G5, Claw Graph K1,3, K3 + 2K1, K2 + 2K2, and Wheel W6.Metelsky, Yu. and Tyshkevich, R. "On Line Graphs of Linear 3-Uniform Hypergraphs." J. Graph Th. 25, 243-251, 1997.Referenced on Wolfram|Alpha
Metelsky GraphsCite this as:
Weisstein, Eric W. "Metelsky Graphs." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MetelskyGraphs.html