The Complexity of Extending Storylines with Minimum Local Crossing Number

Storyline layouts visualize temporal interactions by drawing each character as an $x$-monotone curve and enforcing that the participants of every meeting form a contiguous vertical group. We study a drawing extension variant in which a layout of a sub-storyline is fixed and has to be extended by inserting missing characters while preserving all meeting constraints. We minimize the local crossing number $\chi$, i.e., the maximum number of crossings along any single character. We prove that the problem is W[1]-hard parameterized by the number $k$ of inserted characters plus the maximum number $\sigma$ of active characters, in XP parameterized by $\sigma$ and in FPT parameterized by $\sigma +\chi$.