  An open question in the theory of word-representable graphs for the past
decade has been whether the line graph of a non-word-representable graph is
always non-word-representable. By formulating an appropriate optimization
problem for the decision problem of 3-semi-transitive graphs, we show that the
line graph of a non-word-representable graph can be word-representable. Using
IBM's CPLEX solver, we demonstrate for several known word-representable and
non-word-representable graphs that the line graph of a graph is
3-semi-transitive when there is a solution to the optimization problem. This
results in an example where the line graph of a non-word-representable graph is
both 3-semi-transitive and semi-transitive and thus is word-representable.
