Planar Geometric Storyplan Overview

• Planar geometric storyplan is a model that represents a graph as a series of frames where vertices remain fixed and edges are consistently drawn.
• It emphasizes low visual complexity and mental map preservation by ensuring that persistent elements are displayed unchanged across frames.
• Research in this area investigates various formulations, complexity results, and geometric constraints distinguishing topological from strict straight-line requirements.

Searching arXiv for the main storyplan papers and closely related work. {"query":"(Battista et al., 2022) planar geometric storyplan graph stories small point-sets supporting graph stories", "max_results": 5} {"query":"(Dobler et al., 18 Aug 2025) Geometry Matters in Planar Storyplans", "max_results": 5} {"query":"(Binucci et al., 2022) On the Complexity of the Storyplan Problem", "max_results": 5} {"query":"(Fiala et al., 2023) Outerplanar and Forest Storyplans", "max_results": 5} {"query":"(Borrazzo et al., 2019) Graph Stories in Small Area", "max_results": 5} {"query":"(Binucci et al., 24 Aug 2025) Planar Stories of Graph Drawings: Algorithms and Experiments", "max_results": 5} Planar geometric storyplan denotes a family of models for representing a graph, or a fixed drawing of a graph, as a sequence of frames whose union covers the whole structure while each individual frame remains planar and the representation of persistent objects does not change over time. Across the graph-story, storyplan, and planar-story formulations, the common requirements are low visual complexity in each frame and preservation of the viewer’s mental map; the main technical differences concern whether vertices or edges arrive over time, whether edges may be Jordan arcs or must be straight-line segments, and whether complexity is measured by window size, point-set size, area, or minimum frame size (Battista et al., 2022, Binucci et al., 2022, Binucci et al., 24 Aug 2025).

1. Formal models and basic objects

In the storyplan model of Binucci et al., a storyplan of a graph $G=(V,E)$ is a pair $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$, where $\tau:V\to[n]$ is a bijection giving a total order of the vertices, $i_v=\tau(v)$ is the appearance time of $v$, and $j_v=\max_{u\in N[v]}\tau(u)$ is its last visible time. The lifespan of $v$ is therefore $[i_v,j_v]$, and frame $D_i$ draws the induced subgraph on the visible vertices $V_i=\{v\in V:i_v\le i\le j_v\}$. Each vertex appears exactly once and disappears exactly once; a visible vertex keeps the same point in every frame of its lifespan, and a visible edge keeps the same curve in every frame in which it appears (Binucci et al., 2022).

A closely related formulation is the graph story model $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$0, in which the order $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$1 specifies vertex arrivals, each vertex remains visible for exactly $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$2 time steps, and the visible graph at time $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$3 is the induced subgraph $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$4 on the vertices in the current viewing window. At most $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$5 vertices are visible at any time, but the realization may use $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$6 support points, where $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$7 is the number of extra points available as slack (Battista et al., 2022).

A later edge-centric formulation fixes a straight-line geometric graph $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$8 in advance and seeks a sequence of planar frames $\mathcal S=\langle \tau,\{D_i\}_{i\in[n]}\rangle$9 that always show all vertices at their original positions but only a subset of edges. In this model, the first frame is any planar subset of edges, and each subsequent frame introduces exactly one new edge and deletes exactly those currently displayed edges that cross it. The optimization objective is to maximize the minimum frame size over the whole sequence (Binucci et al., 24 Aug 2025).

2. Persistence, planarity, and the role of geometry

The common mental-map condition is persistence of the common subdrawing. In the graph-story setting, realization requires that for every $\tau:V\to[n]$0, $\tau:V\to[n]$1 is a planar drawing of $\tau:V\to[n]$2, and for every $\tau:V\to[n]$3, the restrictions of $\tau:V\to[n]$4 and $\tau:V\to[n]$5 to the common subgraph $\tau:V\to[n]$6 are identical. This means that a vertex keeps the same point for its entire lifetime and an edge keeps the same Jordan arc for its entire lifetime (Battista et al., 2022).

In the original planar storyplan problem, planarity is topological rather than straight-line: edges are drawn as Jordan arcs, and the persistence condition can be reformulated as existence of a sequence of planar embeddings whose restrictions coincide on common subgraphs. This viewpoint is central to the general complexity results, because it turns frame-to-frame consistency into an embedding-compatibility problem (Binucci et al., 2022).

The geometric variant is stricter. “Geometry Matters in Planar Storyplans” proves that the topological and geometric settings differ: there exists a graph that admits a planar storyplan but no planar geometric storyplan in which each frame is a planar straight-line drawing. The paper constructs a 28-vertex graph with three 4-cycles and eight apex-pairs, gives an 8-frame planar storyplan for the topological version, and then proves that no planar geometric storyplan exists because, for a fixed straight-line drawing of one of the 4-cycles, at most two of its edges are “good” for the required straight-line attachments, while at least six distinct apex-pairs would need such attachments (Dobler et al., 18 Aug 2025).

This establishes that straight-line geometry is not a cosmetic restriction. In the topological model, an edge may be routed as a curve around obstructions; in the geometric model, fixed vertex positions, convex hulls, half-planes, and straight-line visibility create additional global constraints. The separation result therefore shows that “planar” and “planar geometric” storyplans are genuinely different classes (Dobler et al., 18 Aug 2025).

3. Complexity landscape

The decision problem $\tau:V\to[n]$7, which asks whether a graph admits a storyplan for some total vertex order, is NP-complete. The fixed-order variant $\tau:V\to[n]$8, in which the total order is part of the input and one only has to choose the frame drawings, is also NP-complete. The general upper bound proceeds by guessing an order and propagating compatible planar embeddings frame by frame, yielding a $\tau:V\to[n]$9-time exact algorithm and membership in NP (Binucci et al., 2022).

The graph-story realizability problem is likewise hard. For any integer $i_v=\tau(v)$0, testing realizability of $i_v=\tau(v)$1 on $i_v=\tau(v)$2 points is NP-hard; combined with the finite-certificate argument, this gives NP-completeness. At the same time, if $i_v=\tau(v)$3, there is an $i_v=\tau(v)$4-time algorithm, so the problem is fixed-parameter tractable when parameterized by $i_v=\tau(v)$5 (Battista et al., 2022).

The original storyplan problem also admits parameterized algorithms. It is fixed-parameter tractable in the vertex cover number, with running time $i_v=\tau(v)$6, and in the feedback edge set number, with running time $i_v=\tau(v)$7. The second result is based on a linear kernel (Binucci et al., 2022).

For edge-only planar stories of a fixed geometric drawing, the decision version $i_v=\tau(v)$8, asking whether the minimum frame size can be at least $i_v=\tau(v)$9, is NP-complete. The paper also proves that choosing two disjoint independent sets of the crossing graph with sufficiently large minimum size is NP-complete, and that an optimal story may have to start from a planar frame that is not maximal (Binucci et al., 24 Aug 2025).

The geometric recognition problem remains hard as well. “Geometry Matters in Planar Storyplans” adapts the reduction from the topological setting and proves that deciding whether a graph admits a planar geometric storyplan is NP-hard. The paper does not claim NP-completeness; it explicitly leaves open whether the geometric recognition problem is even in NP (Dobler et al., 18 Aug 2025).

4. Positive classes, obstructions, and strict separations

Despite the hardness results, several broad graph classes always admit storyplans. Every partial $v$0-tree admits a storyplan, and such a storyplan can be computed in linear time. This gives a strong positive result for graphs of treewidth at most $v$1 (Binucci et al., 2022).

For stricter per-frame requirements, the landscape becomes more structured. “Outerplanar and Forest Storyplans” proves the strict containment chain

$v$2

Every partial $v$3-tree admits a straight-line outerplanar storyplan in linear time. Every subcubic graph except $v$4 admits a straight-line outerplanar storyplan with at most five edges in each frame, again in linear time. Every triangle-free subcubic graph admits a straight-line forest storyplan with at most five edges per frame, every triangle-free outerplanar graph admits a straight-line forest storyplan in linear time, and every triangle-free planar graph admits a straight-line forest storyplan in polynomial time (Fiala et al., 2023).

The same paper also provides sharp obstructions. No triangulation except $v$5 admits an outerplanar storyplan. There exists a triangle-free 6-regular graph with no planar storyplan, a planar $v$6-free 4-regular graph with no outerplanar storyplan, and a triangle-free 4-regular graph that admits an outerplanar storyplan but no forest storyplan (Fiala et al., 2023).

In the graph-story setting with fixed viewing window, the minimal case $v$7 is already subtle. If $v$8, every minimal graph story is realizable, even if $v$9 is nonplanar. Every minimal graph story with $j_v=\max_{u\in N[v]}\tau(u)$0 outerplanar is realizable, and any outerplanar embedding of $j_v=\max_{u\in N[v]}\tau(u)$1 is a supporting embedding. By contrast, for every $j_v=\max_{u\in N[v]}\tau(u)$2, there exists a minimal graph story with $j_v=\max_{u\in N[v]}\tau(u)$3 series-parallel that is not realizable. For $j_v=\max_{u\in N[v]}\tau(u)$4, every minimal graph story with $j_v=\max_{u\in N[v]}\tau(u)$5 a planar triconnected cubic graph is realizable, and compatible embeddings can be found in $j_v=\max_{u\in N[v]}\tau(u)$6 time (Battista et al., 2022).

These results show that stronger per-frame simplicity does not align monotonically with ordinary planarity of the union graph. A plausible implication is that storyplan feasibility is governed less by the existence of a single global planar embedding than by the interaction between temporal order, lifespan overlap, and the local structures that must become simultaneously visible.

5. Point sets, area bounds, and resource parameters

One major line of work studies how much geometric support a storyplan needs. In the graph-story model, the realizability problem asks whether the sequence can be drawn on only $j_v=\max_{u\in N[v]}\tau(u)$7 points. The case $j_v=\max_{u\in N[v]}\tau(u)$8 is minimal: after the first $j_v=\max_{u\in N[v]}\tau(u)$9 vertices, each new vertex must reuse the point vacated by the vertex that just left. More generally, the paper proves that realizability depends only on the cardinality of the point set, not on its geometry: a graph story is realizable on a set $v$0 of size $v$1 if and only if it is realizable on any other point set $v$2 of the same size. In this topological setting, the exact coordinates do not matter; only the number of support points does (Battista et al., 2022).

The same paper gives strong lower bounds. Some graph stories cannot be drawn for any fixed $v$3, and nested triangles with $v$4 require $v$5. It also proves explicit nonzero lower bounds for certain series-parallel stories: for any $v$6, there exists a series-parallel graph story that is not realizable for

$v$7

Thus extra point slack may be necessary, and in some families no constant amount of slack suffices (Battista et al., 2022).

A complementary resource question is area. “Graph Stories in Small Area” studies straight-line planar drawing stories on the grid. For any path story $v$8, one can compute in $v$9 time a planar straight-line drawing story on a $[i_v,j_v]$0 grid. For any tree story $[i_v,j_v]$1, one can compute in $[i_v,j_v]$2 time a planar straight-line drawing story on an $[i_v,j_v]$3 grid. But for general planar graph stories, area depending only on $[i_v,j_v]$4 is impossible: for a nested-triangles story with $[i_v,j_v]$5, any drawing story lies on an $[i_v,j_v]$6 grid (Borrazzo et al., 2019).

Taken together, these results separate three different geometric resources: point-set cardinality, frame area, and straight-line realizability. In the topological point-set model, geometry collapses to cardinality; in the straight-line area model, geometry reappears through monotonicity, wedges, and nesting obstructions (Battista et al., 2022, Borrazzo et al., 2019).

6. Variants, algorithmic practice, and related formulations

A recent edge-centric variant fixes a geometric graph drawing and asks for a planar story of its edges. Every frame contains all vertices at fixed positions, but only a subset of noncrossing edges. The objective is to maximize the minimum frame size

$[i_v,j_v]$7

The paper models the problem through the crossing graph $[i_v,j_v]$8, where vertices of $[i_v,j_v]$9 correspond to edges of the original drawing and independent sets correspond exactly to planar frames. It gives an exact ILP, several greedy heuristics, and proves that the strongest heuristic is optimal for 2-plane geometric graphs and for those 3-plane geometric graphs whose crossing graph is acyclic (Binucci et al., 24 Aug 2025).

This edge-only model shifts emphasis from induced visible subgraphs to reconfiguration of a fixed drawing. A plausible interpretation is that it studies the same mental-map principle under an even stricter geometric regime: the layout is completely fixed, all vertices are always present, and only edge visibility changes. Its optimization criterion also differs from the classical recognition questions, because it seeks dense frames rather than mere existence (Binucci et al., 24 Aug 2025).

A more distant but conceptually related formulation is Swap Planarity, where the state is a straight-line drawing on a fixed point set and a move swaps the locations of the endpoints of a selected edge. The paper proves that deciding whether an embedded graph can be made planar using swaps is NP-complete, gives $D_i$0 and $D_i$1 bounds on the number of swaps needed, and studies instance generation, order type, and local nonmonotonic progress (Kraaijer et al., 2019). This does not define storyplans in the formal sense used by graph-story and storyplan papers, but it shares the broader theme that planarity restoration can be organized as a sequence of small, visually meaningful steps on fixed geometry.

Across these formulations, planar geometric storyplan is best viewed as a research area rather than a single theorem. Its central technical question is how much of a complex graph can be shown at once, and in what order, when one insists that every visible frame remain planar and that whatever survives from one frame to the next remain unchanged. The accumulated results show that the answer depends sharply on whether one allows Jordan arcs or requires straight segments, whether the temporal process is vertex-based or edge-based, and whether the governing resource is point budget, area, treewidth, or minimum frame density (Battista et al., 2022, Dobler et al., 18 Aug 2025, Binucci et al., 24 Aug 2025).