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 $χ$, 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 $σ$ of active characters, in XP parameterized by $σ$ and in FPT parameterized by $σ+χ$.

翻译：故事线布局通过将每个角色绘制为$x$-单调曲线，并强制要求每次会议的所有参与者构成一个连续的垂直组，从而可视化时间交互。我们研究一种绘图扩展变体：给定子故事线布局已固定，需通过插入缺失角色来扩展该布局，同时满足所有会议约束。我们以最小化局部交叉数$χ$（即任一角色曲线上的最大交叉次数）为目标。我们证明该问题在参数$k$（插入角色数）与$σ$（最大活跃角色数）之和下是W[1]-难的，在参数$σ$下属于XP类，在参数$σ+χ$下属于FPT类。