Fixed-parameter tractability for complements of line graphs

Determine whether computing the boxicity of complements of line graphs is fixed-parameter tractable when parameterized by the boxicity.

Background

Theorem \ref{thm:box_co_line} gives an $O(n^{5k+4})$-time algorithm for deciding whether the complement of a line graph has boxicity at most $k$. Thus, for fixed $k$, the problem is polynomial-time solvable, but the exponent depends on $k$.

The authors pose the parameterized-complexity question of whether this dependence can be essentially improved to fixed-parameter tractability, explicitly conditioning the question on a positive answer to the preceding NP-hardness problem.