Geometry Matters in Planar Storyplans

TL;DR

This paper explores the differences between topological and geometric planar storyplans, showing that some graphs admit topological but not geometric storyplans, and proves recognizing geometric storyplans is NP-hard.

Contribution

It demonstrates that the geometric and topological versions of planar storyplans differ and establishes NP-hardness for recognizing graphs with geometric storyplans.

Findings

Existence of graphs with topological but not geometric planar storyplans.

Recognition of geometric storyplans is NP-hard.

Adapts topological reduction to geometric setting.

Abstract

A storyplan visualizes a graph $G=(V,E)$ as a sequence of $\ell$ frames $\Gamma_1, \dots, \Gamma_\ell$, each of which is a drawing of the induced subgraph $G[V_i]$ of a vertex subset $V_i \subseteq V$. Moreover, each vertex $v \in V$ is contained in a single consecutive sequence of frames $\Gamma_i, \dots, \Gamma_j$, 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.