Complexity of interval completion for line graphs

Determine the computational complexity of the interval completion problem for line graphs.

Background

Interval completion asks for a minimum-edge interval graph containing a given graph as a subgraph. The paper solves this problem for complements of line graphs by enumerating their inclusion-wise minimal interval completions.

The analogous complexity for line graphs themselves is not resolved. The authors motivate the question through the dual relationship between interval completions and covers by interval-order subgraphs, which underlies their boxicity arguments.

References

The latter motivates the following problem.

\begin{problem}\label{prob:completion} What is the complexity of the interval completion problem for line graphs? \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:completion}; page number unavailable in the source