NP-hardness of boxicity for complements of line graphs

Determine whether computing the boxicity of complements of line graphs is NP-hard.

Background

The paper establishes that, for a complement of a line graph, deciding whether the boxicity is at most a fixed integer kk can be performed in polynomial time, placing the bounded-boxicity decision problem in XP when parameterized by kk.

The authors explicitly state that they did not determine whether fixing kk is necessary and leave the computational complexity of unrestricted boxicity computation unresolved. The problem asks specifically whether this computation is NP-hard.

References

We did not manage to show that fixing $k$ is really necessary.

\begin{problem}\label{prob:co_line_NP_hard} Is computing the boxicity of complements of line graphs $NP$-hard? \end{problem}

— On the Boxicity of Line Graphs and of Their Complements  (2501.05049 - Caoduro et al., 9 Jan 2025) in Section 7, Conclusion, Problem \ref{prob:co_line_NP_hard}; page number unavailable in the source