We study the problem of gradually representing a complex graph as a sequence of drawings of small subgraphs whose union is the complex graph. The sequence of drawings is called \emph{storyplan}, and each drawing in the sequence is called a \emph{frame}. In an (outer)planar storyplan, every frame is (outer)planar; in a forest storyplan, every frame is acyclic. It is known that every graph of treewidth at most 3 admits a planar storyplan and that deciding whether a given graph admits a planar storyplan is NP-complete [Binucci et al., JCSS, 2024]. We first prove that deciding whether a given graph admits an outerplanar storyplan (or a forest storyplan) is NP-complete. Then, we show that the FPT algorithms of Binucci et al. also work for our problem variants with small modifications. We identify graph families that admit outerplanar and forest storyplans and families for which such storyplans do not always exist. In the affirmative case, we present efficient algorithms that produce straight-line storyplans.

翻译：我们研究将复杂图逐步表示为一系列小子图绘制的问题，这些子图的并集构成原复杂图。该绘制序列称为\emph{故事平面图}，序列中的每个绘制称为\emph{帧}。在（外）平面故事平面图中，每一帧均为（外）平面图；在森林故事平面图中，每一帧均为无环图。已知树宽不超过3的图均允许平面故事平面图，且判定给定图是否允许平面故事平面图是NP完全问题[Binucci等人，JCSS，2024]。我们首先证明判定给定图是否允许外平面故事平面图（或森林故事平面图）是NP完全的。随后，我们通过对Binucci等人的FPT算法进行微调，证明其同样适用于我们的问题变体。我们识别了允许外平面与森林故事平面图的图族，以及此类故事平面图并非始终存在的图族。在肯定性情形中，我们提出了生成直线绘制故事平面图的高效算法。