Planar Story: Graph & Narrative Visualization

• Planar Story is a representation method that unfolds graphs or narratives into sequential 2D drawings with fixed vertex positions while ensuring planarity.
• It formalizes models such as fixed-window and storyplan formulations to analyze graph evolution, algorithmic complexity, and geometric constraints.
• Planar Story techniques enable dynamic visualizations that bridge graph drawing with narrative diagrams, informing both theoretical and practical applications.

Searching arXiv for papers on planar story / storyplans / graph stories. Planar Story denotes a family of representations in which a graph, storyline, or narrative is unfolded as a sequence of two-dimensional, consistency-preserving views. In graph drawing, the term is formalized through graph stories and storyplans: each frame is a planar drawing of an induced subgraph, vertices remain fixed while they are visible, and the full sequence covers the underlying graph (Borrazzo et al., 2019, Binucci et al., 2022). In adjacent work on narrative visualization and generative storytelling, the same planar intuition organizes entities, events, or scenes on a plane by time, layout, or panel order rather than by a single static image (Bolte et al., 2020, Bae et al., 2024).

1. Core formalizations

The fixed-window formulation begins with a graph $G$, a bijection $\tau:V(G)\to\{1,\dots,n\}$, and a window size $W$. The time-indexed subgraphs are

$$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$

for $t=1,2,\dots,n+W-1$. The full story $\langle G,\tau,W\rangle$ is naturally associated with the sequence $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$ (Borrazzo et al., 2019).

The storyplan formulation replaces the fixed window by vertex lifespans induced by a total order. A storyplan is a pair $\mathcal{S}=\langle \tau,(D_i)_{i\in[n]}\rangle$, where $i_v=\tau(v)$ is the appearance time of $v$ and

$\tau:V(G)\to\{1,\dots,n\}$0

is its lifespan end. A vertex $\tau:V(G)\to\{1,\dots,n\}$1 is visible in every frame $\tau:V(G)\to\{1,\dots,n\}$2. Each drawing $\tau:V(G)\to\{1,\dots,n\}$3 is a planar drawing of the induced subgraph $\tau:V(G)\to\{1,\dots,n\}$4; the point representing a vertex and the curve representing an edge do not change throughout the sequence (Binucci et al., 2022).

A later formulation makes the visibility interval explicit. A storyplan of a graph $\tau:V(G)\to\{1,\dots,n\}$5 on time steps $\tau:V(G)\to\{1,\dots,n\}$6 consists of a pair $\tau:V(G)\to\{1,\dots,n\}$7, where each vertex $\tau:V(G)\to\{1,\dots,n\}$8 has a visible interval $\tau:V(G)\to\{1,\dots,n\}$9, each frame is the drawing $W$0 of the induced subgraph of visible vertices, each vertex is contained in a single consecutive sequence of frames, all vertices and edges contained in consecutive frames are drawn identically, and the union of all frames is a drawing of $W$1 (Dobler et al., 18 Aug 2025).

2. Model variants and design constraints

The literature uses “planar story” for several closely related, but not identical, frame systems.

Model	Core object	Defining constraint
Graph story	Sequence $W$2	Each drawing is straight-line and planar
Small-point-set graph story	Realization on $W$3 given points	Same drawing for a vertex or edge for its entire life
Planar story of a geometric graph	Sequence $W$4	Exactly one new edge enters per frame and crossing edges are deleted

In the small-point-set model, a realization of $W$5 uses only $W$6 fixed points, and the paper shows that “the problem does not depend on the specific set of points but only on its size”; it is NP-hard and is FPT when parameterized by $W$7 (Battista et al., 2022). This model makes the geometric budget itself part of the definition of a planar story.

In the geometric-graph model, a planar story is a sequence of planar frames

$W$8

such that, in the first frame, one visualizes a suitable subset of non-crossing edges, and in each subsequent frame exactly one new edge enters the visualization while all the edges that cross with it are deleted. The objective is to maximize

$W$9

that is, to maximize the minimum frame size (Binucci et al., 24 Aug 2025).

3. Existence results and obstructions

Constructive theory is strongest for paths, trees, outerplanar graphs, triangle-free classes, and bounded-treewidth families. For any path story $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$0, it is possible to compute in $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$1 time a drawing story that is planar, straight-line, and lies on a $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$2 grid. For any tree story $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$3, it is possible to construct in $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$4 time a drawing story that is planar, straight-line, and lies on an $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$5 grid (Borrazzo et al., 2019).

For realizability on small point sets, several sharp positive results are known. Every minimal graph story $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$6 with $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$7 outerplanar is realizable, and any outerplanar embedding of $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$8 is a supporting embedding. Every minimal story $$G_t := G\big[\{v \in V(G) \mid t-W < \tau(v) \le t\}\big],$$9 where $t=1,2,\dots,n+W-1$0 is an $t=1,2,\dots,n+W-1$1-vertex planar triconnected cubic graph is realizable. For $t=1,2,\dots,n+W-1$2, every minimal story $t=1,2,\dots,n+W-1$3 is 1-reroute realizable iff $t=1,2,\dots,n+W-1$4 does not contain $t=1,2,\dots,n+W-1$5 (Battista et al., 2022).

The storyplan literature strengthens these existence results for structurally sparse classes. Every partial 2-tree admits a straight-line outerplanar storyplan, and such a storyplan can be computed in linear time. Every subcubic graph except $t=1,2,\dots,n+W-1$6 admits a straight-line outerplanar storyplan with at most five edges in each frame. Every $t=1,2,\dots,n+W-1$7-free subcubic graph admits a straight-line forest storyplan, and every $t=1,2,\dots,n+W-1$8-free planar graph admits a straight-line forest storyplan (Fiala et al., 2023).

Negative results are equally central. There exist graph stories of planar graphs whose subgraphs cannot be drawn within an area that is only a function of $t=1,2,\dots,n+W-1$9 (Borrazzo et al., 2019). For any $\langle G,\tau,W\rangle$0, there exists a minimal story $\langle G,\tau,W\rangle$1 such that $\langle G,\tau,W\rangle$2 is series-parallel and $\langle G,\tau,W\rangle$3 is not realizable, and for any $\langle G,\tau,W\rangle$4 there exists a story $\langle G,\tau,W\rangle$5 with $\langle G,\tau,W\rangle$6 series-parallel that is not realizable for $\langle G,\tau,W\rangle$7 (Battista et al., 2022). In outerplanar and forest storyplans, no triangulation except $\langle G,\tau,W\rangle$8 admits an outerplanar storyplan; there is a $\langle G,\tau,W\rangle$9-free 6-regular graph that does not admit a planar storyplan; there is a $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$0-free 4-regular planar graph that does not admit an outerplanar storyplan; and there is a $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$1-free 4-regular nonplanar graph that admits an outerplanar storyplan but does not admit a forest storyplan (Fiala et al., 2023).

4. Algorithms, complexity, and geometry

The decision version of StoryPlan is computationally hard. StoryPlan is NP-hard, has no $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$2-time algorithm unless ETH fails, is in NP, and can be solved in $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$3 time. It is FPT when parameterized by the vertex cover number, with running time

$\langle G_1,G_2,\dots,G_{n+W-1}\rangle$4

and FPT when parameterized by the feedback edge set number, with running time

$\langle G_1,G_2,\dots,G_{n+W-1}\rangle$5

The same work also proves that partial 3-trees always admit a storyplan, computable in linear time, and that for planar graphs the framewidth equals the pathwidth (Binucci et al., 2022).

For fixed point-set graph stories, the realizability problem is NP-complete, but it is FPT when parameterized by $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$6. This places small-point planar stories in the same pattern as StoryPlan proper: hard in general, but tractable under strong structural restrictions (Battista et al., 2022).

In the geometric-graph setting, MaxMinFramePlanarStoryD is NP-complete. Still, there are exact algorithms on restricted classes: for 2-plane geometric graphs, MaxMinFramePlanarStory can be solved optimally in linear time; for 3-plane geometric graphs whose crossing graph has no cycles, it can be solved optimally in cubic time. The paper also gives an ILP formulation and six heuristic variants $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$7, with $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$8 giving the highest quality and $\langle G_1,G_2,\dots,G_{n+W-1}\rangle$9 the best efficiency/effectiveness trade-off (Binucci et al., 24 Aug 2025).

A further refinement distinguishes topological from geometric storyplans. There exists a graph that admits a planar storyplan, but no planar geometric storyplan, in which each frame is a planar straight-line drawing; recognizing the graphs that admit planar geometric storyplans remains NP-hard (Dobler et al., 18 Aug 2025). This establishes that geometry matters in planar storyplans: the topological and geometric settings do not coincide.

5. Planar narrative diagrams beyond frame sequences

A second branch of the literature interprets planar story not as a sequence of induced subgraphs but as a single two-dimensional narrative diagram. Storyline visualizations are planar (2D) diagrams that place time along one axis and represent entities as lines or “streams” that move through time, showing when they coexist, meet, split, or disappear (Bolte et al., 2020).

Organic Narrative Charts extend this tradition by explicitly supporting links between entities, splits and merges of streams, and nesting or hierarchies of streams. Streams are defined over time intervals $\mathcal{S}=\langle \tau,(D_i)_{i\in[n]}\rangle$0, thickness varies over time, labels can be inside, outside, or on top, and the layout is produced by a force-directed algorithm with fixed time coordinates, repulsive forces, edge-specific forces, and nesting constraints. The rendering deliberately imitates Ward Shelley’s diagrammatic paintings through cubic Bézier curves, textured fills via fractal noise, black outlines, and SVG-based filters. The paper also provides an open-source authoring tool prototype (Bolte et al., 2020).

This suggests a broader meaning of planar story in visualization research: not only a sequence of planar frames, but also a planar surface on which temporal structure, grouping, influence, and annotation are made simultaneously visible.

6. Planning, premise design, and story-to-frame generation

A plausible extension of the term appears in recent generative work, where a story is treated as a structured sequence of scenes, plans, or modules rather than as a single prompt. The common move is decomposition: a long narrative is broken into explicit, editable units that can be ordered, revised, and rendered.

Neural Story Planning unifies causal planning with neural LLMs by using commonsense knowledge extracted from LLMs to recursively expand a story plot in a backward chaining fashion; results indicate that the method produces more coherent plotlines than several strong baselines (Ye et al., 2022). CritiCS introduces a plan refining stage (CrPlan) and a story generation stage (CrText), in which a group of LLM critics and one leader collaborate to incrementally refine drafts of plan and story throughout multiple rounds; extensive human evaluation shows that the framework can significantly enhance story creativity and reader engagement, while also maintaining narrative coherence (Bae et al., 2024).

MoPS, or Modular Story Premise Synthesis, breaks down story premises into modules like background and persona and proceeds in three phases: precollect a consistent set of candidates for each module to form a nested dictionary, extract a key path from the nested dictionary as the premise design, and instruct an LLM to integrate the design into a coherent premise sentence. It reports 7.6k generated premises and 1k extended stories (Ma et al., 2024). In story illustration, S2ED proposes Story-to-Executable Descriptions, a training-free, model-agnostic, prompt-layer framework that converts a full story into a sequence of explicit, editable executable descriptions for more consistent rendering, improves sequence-level consistency and character fidelity on Flintstones and Shakoo Maku, and is deployed in an end-to-end story-to-storybook system (Yin et al., 21 May 2026).

Taken together, these systems suggest a contemporary, generative reading of planar story: a story is flattened into a sequence of stable, local, inspectable representations—frames, plans, panels, or executable descriptions—so that coherence can be enforced over time, not merely at the level of a single output.