Polynomial-time recognition of line graphs with boxicity two

Determine whether it can be decided in polynomial time whether a given line graph has boxicity at most \(2\).

Background

The paper explains that the computational complexity of boxicity for line graphs is less understood than for complements of line graphs: even membership in XP is open. It isolates the decision problem for boxicity at most two as a concrete unresolved case.

The authors observe that $\boxi(L(K_4))=3$, so the open problem only concerns line graphs of underlying graphs whose clique number is at most three.

References

Problem \ref{prob:co_line_NP_hard} arises for line graphs as well, where even the membership to the class $XP$ is open:

\begin{problem}\label{prob:line_graph} Can it be decided in polynomial time whether the boxicity of a line graph is $2$? \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:line_graph}; page number unavailable in the source