  Given a graph G, its triangular line graph is the graph T(G) with vertex set
consisting of the edges of G and adjacencies between edges that are incident in
G as well as being within a common triangle. Graphs with a representation as
the triangular line graph of some graph G are triangular line graphs, which
have been studied under many names including anti-Gallai graphs, 2-in-3 graphs,
and link graphs. While closely related to line graphs, triangular line graphs
have been difficult to understand and characterize. Van Bang Le asked if
recognizing triangular line graphs has an efficient algorithm or is
computationally complex. We answer this question by proving that the complexity
of recognizing triangular line graphs is NP-complete via a reduction from
3-SAT.
