Geometry Matters in Planar Storyplans

A storyplan visualizes a graph $G=(V,E)$ as a sequence of frames Γ₁, , Γ_, each of which is a drawing of the induced subgraph G[Vᵢ] of a vertex subset Vᵢ ⊆ V. Moreover, each vertex v ∈ V is contained in a single consecutive sequence of frames Γᵢ, , Γⱼ, all vertices and edges contained in consecutive frames are drawn identically, and the union of all frames is a drawing of G. In GD 2022, the concept of planar storyplans was introduced, in which each frame must be a planar (topological) drawing. Several (parameterized) complexity results for recognizing graphs that admit a planar storyplan were provided, including NP-hardness. In this paper, we investigate an open question posed in the GD paper and show that the geometric and topological settings of the planar storyplan problem differ: We provide an instance of a graph that admits a planar storyplan, but no planar geometric storyplan, in which each frame is a planar straight-line drawing. Still, by adapting the reduction proof from the topological to the geometric setting, we show that recognizing the graphs that admit planar geometric storyplans remains NP-hard.