On the Complexity of the Storyplan Problem

Abstract

Motivated by dynamic graph visualization, we study the problem of representing a graph $G$ in the form of a \emph{storyplan}, that is, a sequence of frames with the following properties. Each frame is a planar drawing of the subgraph of $G$ induced by a suitably defined subset of its vertices. Between two consecutive frames, a new vertex appears while some other vertices may disappear, namely those whose incident edges have already been drawn in at least one frame. In a storyplan, each vertex appears and disappears exactly once. For a vertex (edge) visible in a sequence of consecutive frames, the point (curve) representing it does not change throughout the sequence. Note that the order in which the vertices of $G$ appear in the sequence of frames is a total order. In the \textsc{StoryPlan} problem, we are given a graph and we want to decide whether there exists a total order of its vertices for which a storyplan exists. We prove that the problem is NP-complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of $G$. Also, we prove that partial $3$-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP-complete in the case in which the total order of the vertices is given as part of the input and we have to choose how to draw the frames.