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Outerplanar and Forest Storyplans

Jiří Fiala, Oksana Firman, Giuseppe Liotta, Alexander Wolff, Johannes Zink

TL;DR

This paper extends the notion of planar storygraphs by introducing outerplanar and forest storyplans, and establishes foundational complexity and algorithmic results for these variants. It proves NP-hardness (and NP membership) for deciding whether a graph admits an outerplanar or forest storyplan, adapting the reduction framework from Planar Storyplan via variable and clause gadgets. It further shows that the FPT algorithms developed for Planar Storyplan adapt to these variants, and it identifies graph classes that admit or forbid such storyplans, including explicit constructions and negative results for triangulations and certain regular graphs. When feasible, it provides efficient, straight-line storyplan constructions and a vertex-cover–based kernelization to achieve FPT performance, thereby clarifying the landscape of tractability across graph classes and parameter regimes.

Abstract

We study the problem of gradually representing a complex graph as a sequence of drawings of small subgraphs whose union is the complex graph. The sequence of drawings is called \emph{storyplan}, and each drawing in the sequence is called a \emph{frame}. In an (outer)planar storyplan, every frame is (outer)planar; in a forest storyplan, every frame is acyclic. It is known that every graph of treewidth at most 3 admits a planar storyplan and that deciding whether a given graph admits a planar storyplan is NP-complete [Binucci et al., JCSS, 2024]. We first prove that deciding whether a given graph admits an outerplanar storyplan (or a forest storyplan) is NP-complete. Then, we show that the FPT algorithms of Binucci et al. also work for our problem variants with small modifications. We identify graph families that admit outerplanar and forest storyplans and families for which such storyplans do not always exist. In the affirmative case, we present efficient algorithms that produce straight-line storyplans.

Outerplanar and Forest Storyplans

TL;DR

This paper extends the notion of planar storygraphs by introducing outerplanar and forest storyplans, and establishes foundational complexity and algorithmic results for these variants. It proves NP-hardness (and NP membership) for deciding whether a graph admits an outerplanar or forest storyplan, adapting the reduction framework from Planar Storyplan via variable and clause gadgets. It further shows that the FPT algorithms developed for Planar Storyplan adapt to these variants, and it identifies graph classes that admit or forbid such storyplans, including explicit constructions and negative results for triangulations and certain regular graphs. When feasible, it provides efficient, straight-line storyplan constructions and a vertex-cover–based kernelization to achieve FPT performance, thereby clarifying the landscape of tractability across graph classes and parameter regimes.

Abstract

We study the problem of gradually representing a complex graph as a sequence of drawings of small subgraphs whose union is the complex graph. The sequence of drawings is called \emph{storyplan}, and each drawing in the sequence is called a \emph{frame}. In an (outer)planar storyplan, every frame is (outer)planar; in a forest storyplan, every frame is acyclic. It is known that every graph of treewidth at most 3 admits a planar storyplan and that deciding whether a given graph admits a planar storyplan is NP-complete [Binucci et al., JCSS, 2024]. We first prove that deciding whether a given graph admits an outerplanar storyplan (or a forest storyplan) is NP-complete. Then, we show that the FPT algorithms of Binucci et al. also work for our problem variants with small modifications. We identify graph families that admit outerplanar and forest storyplans and families for which such storyplans do not always exist. In the affirmative case, we present efficient algorithms that produce straight-line storyplans.
Paper Structure (9 sections, 7 theorems, 8 figures)

This paper contains 9 sections, 7 theorems, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Separation of Graph Classes
  4. Computational Complexity
  5. Variable gadget.
  6. Clause gadget.
  7. Correctness.
  8. Parameterized Algorithms
  9. Algorithm with Respect to the Vertex Cover Number

Key Result

Lemma 3

Let $K_{a,b} = (A \cup B, E)$ be a complete bipartite graph with $a = |A|$, $b=|B|$, and $3 \le a \le b$. Let $\mathcal{S}=\langle \tau, \{D_i\}_{i\in [a+b]} \rangle$ be a planar storyplan of $K_{a,b}$. Exactly one of $A$ and $B$ is such that all its vertices are visible in some frame $i \in [a+b]$.

Figures (8)

  • Figure 1: A forest storyplan of the Petersen graph.
  • Figure 2: Overview: existing bdllmms-csp-JCSS24 and new storyplan results, implying $\mathcal{G}_\mathrm{forest} \subsetneq \mathcal{G}_\mathrm{outerpl} \subsetneq \mathcal{G}_\mathrm{planar} \subsetneq \mathcal{G}$. (For simplicity, we mention 2-/3-trees rather than partial 2-/3-trees.)
  • Figure 3: The dodecahedron with a vertex numbering that corresponds to a forest storyplan.
  • Figure 4: Three graphs from the proof of \ref{['thm:negative']}. The graph in (\ref{['fig:triangle-free1']}) is $\triangle$-free and does not admit any planar storyplan. The octahedron graph in (\ref{['fig:octahedron']}) does not admit any outerplanar storyplan. The graph in (\ref{['fig:triangle-free3']}) is $\triangle$-free and does not admit any forest storyplan (but the vertex numbering corresponds to an outerplanar storyplan -- if vertex 8 is placed at the position of vertex 6, which will have disappeared by then).
  • Figure 5: The gadget $H_i=K_{18,3}$ for a variable $x_i$ that occurs three times in $F$.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 1
  • Lemma 3: bdllmms-csp-JCSS24
  • Example 4: Bipartite graphs
  • Lemma 5
  • Theorem 6
  • Example 7: Platonic graphs
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • Theorem 11