Complexity class of recognizing planar geometric storyplans

Determine whether the decision problem that asks, given a graph G, whether G admits a planar geometric storyplan (a sequence of frames consisting of planar straight-line drawings satisfying the storyplan visibility and consistency conditions) is in NP, or whether it is complete for another complexity class such as ∃ℝ.

Background

Planar storyplans represent graphs as sequences of planar frames with consistency constraints across frames. This paper shows that some graphs admitting planar (topological) storyplans do not admit planar geometric (straight-line) storyplans, and it proves that recognizing graphs that admit planar geometric storyplans is NP-hard by adapting a reduction from One-In-3SAT.

While NP-hardness is established, the authors do not resolve whether the geometric recognition problem is in NP. Many geometric decision problems fall into complexity classes beyond NP, such as ∃ℝ, motivating the question of a precise complexity classification.