Planar Storyplan: Dynamic Graph Visualization

• Planar storyplan is a dynamic graph visualization model where vertices appear once and persist with unchanging geometry across ordered planar snapshots.
• It enforces temporal stability through a strict total ordering and visibility intervals, ensuring each frame is a planar drawing of the induced subgraph.
• The NP-hard decision problem has led to parameterized algorithms for restricted cases, with efficient solutions for classes like partial 3-trees.

Planar storyplan denotes a graph-visualization model in which a graph $G$ is represented by a time-ordered sequence of planar frames, with a total order on vertex appearances and a strict persistence rule: whenever a vertex or edge remains visible across consecutive frames, its geometric representation is unchanged. In the formalization introduced for dynamic graph visualization, each vertex appears exactly once and disappears exactly once, each frame is a planar drawing of an induced subgraph, and the central decision problem asks whether some total order of the vertices admits such a representation (Binucci et al., 2022).

1. Formal model

In the 2022 formalization, a storyplan of a finite graph $G=(V,E)$, with $|V|=n$, is a pair

$\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$

where $\tau:V\rightarrow[n]$ is a bijection giving a total order of the vertices, and $D_i$ is the $i$-th frame (Binucci et al., 2022). If $\tau(v)=i_v$, then $v$ appears at step $i_v$. With the closed neighborhood

$G=(V,E)$0

the endpoint of the visibility interval of $G=(V,E)$1 is

$G=(V,E)$2

The lifespan of $G=(V,E)$3 is therefore $G=(V,E)$4: $G=(V,E)$5 is visible at every step $G=(V,E)$6 and disappears at step $G=(V,E)$7 (Binucci et al., 2022).

For each time $G=(V,E)$8, the visible vertex set is

$G=(V,E)$9

and the visible edge set is

$|V|=n$0

The frame $|V|=n$1 is a planar drawing of the induced subgraph $|V|=n$2. The model imposes two stability constraints. First, 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. Second, if a vertex or edge is visible in a sequence of consecutive frames, the point or curve representing it does not change throughout that sequence (Binucci et al., 2022). The resulting object is neither a general animation nor a sequence of unrelated redrawings; it is a temporally constrained decomposition of a single graph into planar induced snapshots.

2. Decision problem and complexity landscape

The associated decision problem, \textsc{StoryPlan}, asks whether a given graph admits a total order of its vertices for which a storyplan exists (Binucci et al., 2022). This recognition problem is NP-complete. The hardness persists even in the variant in which the total order is part of the input and only the choice of the drawings of the frames remains open (Binucci et al., 2022).

The same work complements hardness with positive algorithmic results. It gives two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of $|V|=n$3. It also identifies a graph class for which realizability is unconditional: partial $|V|=n$4-trees always admit a storyplan, and such a storyplan can be computed in linear time (Binucci et al., 2022). These results place planar storyplans in a characteristic algorithmic position: general recognition is intractable, but structurally restricted instances admit efficient treatment.

A plausible implication is that the difficulty of the model is driven less by planarity of individual frames than by the interaction between three global constraints: induced visibility sets, single-interval lifespans, and the requirement that geometry remain static over each visible interval.

3. Topological versus geometric storyplans

Subsequent work sharpened the notion by separating topological and geometric realizability. In the 2025 paper "Geometry Matters in Planar Storyplans," a storyplan is given by an interval assignment $|V|=n$5 for each vertex and a drawing function $|V|=n$6, with the condition that adjacent vertices have intersecting intervals and that each frame graph

$|V|=n$7

is drawn consistently across time (Dobler et al., 18 Aug 2025). In this formulation, a planar storyplan requires each frame to be planar in the topological sense, whereas a planar geometric storyplan requires each frame to be a planar straight-line drawing (Dobler et al., 18 Aug 2025).

The paper resolves an open question from the earlier work by showing that the two settings differ: there exists a graph that admits a planar storyplan but no planar geometric storyplan (Dobler et al., 18 Aug 2025). At the same time, the negative complexity picture remains: by adapting the reduction proof from the topological to the geometric setting, the paper shows that recognizing graphs that admit planar geometric storyplans is NP-hard (Dobler et al., 18 Aug 2025).

This separation is conceptually important. It shows that the existence of a topologically planar temporal decomposition does not imply the existence of a straight-line realization with the same temporal constraints. In other words, geometry is not a superficial implementation detail; it changes the feasible set.

4. Relation to graph stories and bounded point sets

Planar storyplans belong to a broader family of dynamic graph models that trade global readability against local temporal consistency. In "Graph Stories in Small Area," a graph story is determined by a graph $|V|=n$8, a bijective appearance order $|V|=n$9, and a fixed window size $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$0. The visible graph at time $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$1 is

$\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$2

and a drawing story requires straight-line planar drawings in which every vertex keeps the same coordinates, and every edge keeps the same segment, throughout its lifetime (Borrazzo et al., 2019). The paper gives constructive linear-time algorithms showing that graph stories of paths and trees can be drawn on a $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$3 and an $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$4 grid, respectively, while also proving that there exist planar graph stories whose subgraphs cannot be drawn within an area that is only a function of $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$5 (Borrazzo et al., 2019).

A more combinatorial variant appears in "Small Point-Sets Supporting Graph Stories," where a graph story is a tuple

$\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$6

drawn on a fixed point set $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$7 with $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$8, under planarity and strict mental-map preservation constraints denoted R1 and R2 (Battista et al., 2022). Here realizability depends only on the size of the point set, not on the specific geometry of the points. The paper proves NP-hardness for every fixed $\mathcal{S}=\langle \tau,\{D_i\}_{i\in[n]}\rangle,$9, gives an FPT algorithm parameterized by $\tau:V\rightarrow[n]$0 with running time

$\tau:V\rightarrow[n]$1

and derives a detailed class-sensitive landscape: every minimal graph story with outerplanar $\tau:V\rightarrow[n]$2 is realizable; for any $\tau:V\rightarrow[n]$3 there exist minimal series-parallel stories that are not realizable; every minimal graph story $\tau:V\rightarrow[n]$4 is $\tau:V\rightarrow[n]$5-reroute realizable if and only if $\tau:V\rightarrow[n]$6 does not contain $\tau:V\rightarrow[n]$7; and some families, such as nested triangles, require $\tau:V\rightarrow[n]$8 (Battista et al., 2022).

These models are not identical to \textsc{StoryPlan}, but they share the same core design doctrine: planarity is enforced per frame, and temporal continuity is enforced by keeping geometry fixed while visible.

5. Edge-centric planar stories

A distinct but closely related line of work studies temporal decompositions of a fixed geometric drawing by varying only the edge set. In "Planar Stories of Graph Drawings: Algorithms and Experiments," the input is a geometric graph $\tau:V\rightarrow[n]$9 with fixed vertex coordinates and straight-line edges, possibly with crossings. A planar frame is any subgraph containing all vertices and a subset of pairwise non-crossing edges, and a planar story is a sequence

$D_i$0

such that each edge appears in a non-empty consecutive block of frames, and each frame after the first is obtained by inserting exactly one new edge and deleting exactly those current edges that cross it (Binucci et al., 24 Aug 2025).

The optimization target in that model is the minimum frame size

$D_i$1

with the goal of maximizing $D_i$2. The decision version is NP-complete. The paper also develops exact algorithms for $D_i$3-plane geometric graphs and for certain $D_i$4-plane cases, an ILP formulation for the general problem, and several greedy heuristics based on the crossing graph, where planar frames correspond to independent sets (Binucci et al., 24 Aug 2025).

This model is edge-centric rather than vertex-centric, but it addresses the same two requirements stated explicitly in the paper: drawing stability and low visual complexity (Binucci et al., 24 Aug 2025). It can therefore be read as a neighboring formalism rather than as a reformulation of planar storyplans.

6. Terminological divergence

The phrase "planar storyplan" also appears in a different sense in virtual-reality presentation research. In "Planar or Spatial: Exploring Design Aspects and Challenges for Presentations in Virtual Reality with No-coding Interface," a planar storyplan is defined as a VR presentation whose narrative structure, layout, and navigation mostly follow the logic of $D_i$5D slide-based presentations: focused panels, linear or lightly branched sequences, and minimal required locomotion, embedded in a VR environment that may add selective spatial elements (Wu et al., 20 Oct 2025).

In that usage, the defining properties are discrete stages, $D_i$6D panels, predominantly linear next/previous navigation, and a preference for stationary viewing. The reported advantages are learnability, narrative clarity, accessibility, and efficiency, while the reported limitations are that such presentations may underutilize VR’s spatial affordances and may feel only marginally better than $D_i$7D formats (Wu et al., 20 Oct 2025). This usage describes presentation design rather than graph drawing.

The coexistence of these meanings suggests that "planar storyplan" is a polysemous term across subfields. In graph drawing and dynamic graph visualization, however, its technically established sense is the vertex-ordered sequence of planar induced frames formalized by \textsc{StoryPlan} and extended by later work on geometry, point-set realizability, and edge-centric planar stories (Binucci et al., 2022).