36 Search Results for "Da Lozzo, Giordano"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.36
Upward Book Embeddings of Partitioned Digraphs

Authors: Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph G = (V, ⋃^k_{i=1} E_i), that is, a digraph whose edge set is partitioned into k subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput. 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for k = 1 and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for k ≥ 3. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case k = 2. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when k = 2, thus closing the complexity gap for the problem. Second, we show that, for an n-vertex partitioned digraph with a prescribed planar embedding, the existence of an upward book embedding that respects the given planar embedding can be tested in O(n log³ n) time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial 2-trees.
Cite as

Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter. Upward Book Embeddings of Partitioned Digraphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dalozzo_et_al:LIPIcs.SoCG.2026.36,
  author =	{Da Lozzo, Giordano and Frati, Fabrizio and Rutter, Ignaz},
  title =	{{Upward Book Embeddings of Partitioned Digraphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.36},
  URN =		{urn:nbn:de:0030-drops-258424},
  doi =		{10.4230/LIPIcs.SoCG.2026.36},
  annote =	{Keywords: upward book embeddings, partitioned digraphs, SPQ-trees, 2-trees}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.25
Structural Parameterizations of Simultaneous Planarity

Authors: Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Matthias Pfretzschner, and Ignaz Rutter

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
Given a set of graphs on the same vertex set, the problem Simultaneous Embedding With Fixed Edges (SEFE) asks, whether there exist planar drawings of all input graphs, such that every pair of drawings coincides on their shared subgraph. It is known that SEFE is NP-complete [Elisabeth Gassner et al., 2006], even in the so-called sunflower case, where all pairs of input graphs have the same shared graph G_∩ [Marcus Schaefer, 2012]. Fink, Pfretzschner, and Rutter [Simon D. Fink et al., 2023] recently initiated the study of the parameterized complexity of SEFE in the sunflower case, mainly focusing on structural parameters of G_∩. In this work, we shift the focus towards parameters of the union graph G_∪ that contains the edges of all input graphs. On the positive side, we establish fixed-parameter tractability for the problem with respect to the feedback edge set number of G_∪. We complement this result by showing that it, surprisingly, remains NP-complete even if G_∪ has constant vertex cover number. These results settle two open questions posed by Fink et al. [Simon D. Fink et al., 2023].
Cite as

Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Matthias Pfretzschner, and Ignaz Rutter. Structural Parameterizations of Simultaneous Planarity. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{depian_et_al:LIPIcs.ISAAC.2025.25,
  author =	{Depian, Thomas and Fink, Simon D. and Firbas, Alexander and Ganian, Robert and Pfretzschner, Matthias and Rutter, Ignaz},
  title =	{{Structural Parameterizations of Simultaneous Planarity}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.25},
  URN =		{urn:nbn:de:0030-drops-249332},
  doi =		{10.4230/LIPIcs.ISAAC.2025.25},
  annote =	{Keywords: SEFE, Simultaneous Planarity, Fixed-Parameter Tractability, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.19
Crossing and Non-Crossing Families

Authors: Todor Antić, Martin Balko, and Birgit Vogtenhuber

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
For a finite set P of points in the plane in general position, a crossing family of size k in P is a collection of k line segments with endpoints in P that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of n points in the plane in general position. It is widely believed that this size should be linear in n. Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets P that do not contain a non-crossing family of size m, which is a collection of 4 disjoint subsets P₁, P₂, P₃, and P₄ of P, each containing m points of P, such that for every choice of 4 points p_i ∈ P_i, the set {p₁,p₂,p₃,p₄} is such that p₄ is in the interior of the triangle formed by p₁,p₂,p₃. We prove that, for every m ∈ ℕ, each set P of n points in the plane in general position contains either a crossing family of size n/2^{O(√{log{m}})} or a non-crossing family of size m, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time O(nm^{1+o(1)}). We also prove that a crossing family of size Ω(n/m) or a non-crossing family of size m in P can be found in expected time O(n).
Cite as

Todor Antić, Martin Balko, and Birgit Vogtenhuber. Crossing and Non-Crossing Families. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{antic_et_al:LIPIcs.GD.2025.19,
  author =	{Anti\'{c}, Todor and Balko, Martin and Vogtenhuber, Birgit},
  title =	{{Crossing and Non-Crossing Families}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{19:1--19:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.19},
  URN =		{urn:nbn:de:0030-drops-250058},
  doi =		{10.4230/LIPIcs.GD.2025.19},
  annote =	{Keywords: crossing family, non-crossing family, geometric graph}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.27
Geometry Matters in Planar Storyplans

Authors: Alexander Dobler, Maximilian Holzmüller, and Martin Nöllenburg

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
A storyplan visualizes a graph G = (V,E) as a sequence of 𝓁 frames Γ₁, … , Γ_𝓁, each of which is a drawing of the induced subgraph G[V_i] of a vertex subset V_i ⊆ V. Moreover, each vertex v ∈ V is contained in a single consecutive sequence of frames Γ_i, … , Γ_j, all vertices and edges contained in consecutive frames are drawn identically, and the union of all frames is a drawing of G. In GD 2022, the concept of planar storyplans was introduced, in which each frame must be a planar (topological) drawing. Several (parameterized) complexity results for recognizing graphs that admit a planar storyplan were provided, including NP-hardness. In this paper, we investigate an open question posed in the GD paper and show that the geometric and topological settings of the planar storyplan problem differ: We provide an instance of a graph that admits a planar storyplan, but no planar geometric storyplan, in which each frame is a planar straight-line drawing. Still, by adapting the reduction proof from the topological to the geometric setting, we show that recognizing the graphs that admit planar geometric storyplans remains NP-hard.
Cite as

Alexander Dobler, Maximilian Holzmüller, and Martin Nöllenburg. Geometry Matters in Planar Storyplans. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 27:1-27:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dobler_et_al:LIPIcs.GD.2025.27,
  author =	{Dobler, Alexander and Holzm\"{u}ller, Maximilian and N\"{o}llenburg, Martin},
  title =	{{Geometry Matters in Planar Storyplans}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{27:1--27:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.27},
  URN =		{urn:nbn:de:0030-drops-250135},
  doi =		{10.4230/LIPIcs.GD.2025.27},
  annote =	{Keywords: geometric storyplan, planarity, straight-line drawing, dynamic graph drawing}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.18
Internally-Convex Drawings of Outerplanar Graphs in Small Area

Authors: Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Giuseppe Liotta, and Antonios Symvonis

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
A well-known result by Kant [Algorithmica, 1996] implies that n-vertex outerplane graphs admit embedding-preserving planar straight-line grid drawings where the internal faces are convex polygons in O(n²) area. In this paper, we present an algorithm to compute such drawings in O(n¹·⁵) area. We also consider outerplanar drawings in which the internal faces are required to be strictly-convex polygons. In this setting, we consider outerplanar graphs whose weak dual is a path and give a drawing algorithm that achieves Θ(nk²) area, where k is the maximum size of an internal facial cycle.
Cite as

Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Giuseppe Liotta, and Antonios Symvonis. Internally-Convex Drawings of Outerplanar Graphs in Small Area. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bekos_et_al:LIPIcs.GD.2025.18,
  author =	{Bekos, Michael A. and Da Lozzo, Giordano and Frati, Fabrizio and Liotta, Giuseppe and Symvonis, Antonios},
  title =	{{Internally-Convex Drawings of Outerplanar Graphs in Small Area}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.18},
  URN =		{urn:nbn:de:0030-drops-250042},
  doi =		{10.4230/LIPIcs.GD.2025.18},
  annote =	{Keywords: Grid drawings, convexity, area bounds, outerplanar graphs}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.14
OOPS: Optimized One-Planarity Solver via SAT

Authors: Sergey Pupyrev

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
We present OOPS (Optimized One-Planarity Solver), a practical heuristic for recognizing 1-planar graphs and several important subclasses. A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once - a natural generalization of planar graphs that has received increasing attention in graph drawing and beyond-planar graph theory. Although testing planarity can be done in linear time, recognizing 1-planar graphs is NP-complete, making effective practical algorithms especially valuable. The core idea of our approach is to reduce the recognition of 1-planarity to a propositional satisfiability (SAT) instance, enabling the use of modern SAT solvers to efficiently explore the search space. Despite the inherent complexity of the problem, our method is substantially faster in practice than naïve or brute-force algorithms. In addition to demonstrating the empirical performance of our solver on synthetic and real-world instances, we show how OOPS can be used as a discovery tool in theoretical graph theory. Specifically, we employ OOPS to investigate two research problems concerning 1-planarity of specific graph families. Our implementation of the algorithm is publicly available to support further exploration in the field.
Cite as

Sergey Pupyrev. OOPS: Optimized One-Planarity Solver via SAT. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{pupyrev:LIPIcs.GD.2025.14,
  author =	{Pupyrev, Sergey},
  title =	{{OOPS: Optimized One-Planarity Solver via SAT}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.14},
  URN =		{urn:nbn:de:0030-drops-250004},
  doi =		{10.4230/LIPIcs.GD.2025.14},
  annote =	{Keywords: beyond planarity, 1-planar graph, SAT, book embeddings, upward 1-planarity}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.9
The Page Number of Monotone Directed Acyclic Outerplanar Graphs Is Four or Five

Authors: Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, and Michael Kaufmann

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
A k-page book embedding of a directed acyclic graph consists of a topological order of its vertices and a k-coloring of its edges, such that no two edges of the same color cross, that is, their endpoints do not alternate in the order. The minimum value of k for which such an embedding exists is referred to as the page number of the graph. In contrast to general directed acyclic planar graphs, which may have unbounded page number [SIAM J. Comput. 28(5), 1999], it was recently shown that directed acyclic outerplanar graphs have bounded page number. In particular, Jungeblut, Merker and Ueckerdt provided an upper bound of 24,776 on their page number [FOCS 2023: 1937-1952]. In this work, we focus on so-called monotone directed acyclic outerplanar graphs. Starting from a single edge, these graphs are constructed by iteratively connecting a new vertex to the endpoints of an existing edge on the outer face using either two incoming or two outgoing edges incident to it. These graphs have twist-number 4 [GD 2023: 135-151] (i.e., they admit a topological order in which no more than four edges pairwise cross), a property, which was leveraged by Jungeblut, Merker and Ueckerdt to show that their page number is at most 128. We lower this upper bound to 5 and we also provide a lower bound of 4. A notable consequence of our result is a significant improvement of the upper bound on the page number of general directed outerplanar graphs from 24,776 to 1,160.
Cite as

Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, and Michael Kaufmann. The Page Number of Monotone Directed Acyclic Outerplanar Graphs Is Four or Five. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alam_et_al:LIPIcs.GD.2025.9,
  author =	{Alam, Jawaherul Md. and Bekos, Michael A. and Gronemann, Martin and Kaufmann, Michael},
  title =	{{The Page Number of Monotone Directed Acyclic Outerplanar Graphs Is Four or Five}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.9},
  URN =		{urn:nbn:de:0030-drops-249952},
  doi =		{10.4230/LIPIcs.GD.2025.9},
  annote =	{Keywords: Book embeddings, page number, directed outerplanar graphs}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.4
Heuristics for Exact 1-Planarity Testing

Authors: Simon D. Fink, Miriam Münch, Matthias Pfretzschner, and Ignaz Rutter

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
Since many real-world graphs are nonplanar, the study of graphs that allow few crossings per edge has been an active subfield of graph theory in recent years. One of the most natural generalizations of planar graphs are the so-called 1-planar graphs that admit a drawing with at most one crossing per edge. Unfortunately, testing whether a graph is 1-planar is known to be NP-complete even for very restricted graph classes. On the positive side, Binucci, Didimo and Montecchiani [Binucci et al., 2023] presented the first practical algorithm for testing 1-planarity based on an easy-to-implement backtracking strategy. We build on this idea and systematically explore the design choices of such algorithms and propose several new ingredients, such as different branching strategies and multiple filter criteria that allow us to reject certain branches in the search tree early on. We conduct an extensive experimental evaluation that evaluates the efficiency and effectiveness of these ingredients. Given a time limit of three hours per instance, our best configuration is able to solve more than 95% of the non-planar instances from the well-known North and Rome graphs with up to 50 vertices. Notably, the median running time for solved instances is well below 4 seconds.
Cite as

Simon D. Fink, Miriam Münch, Matthias Pfretzschner, and Ignaz Rutter. Heuristics for Exact 1-Planarity Testing. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fink_et_al:LIPIcs.GD.2025.4,
  author =	{Fink, Simon D. and M\"{u}nch, Miriam and Pfretzschner, Matthias and Rutter, Ignaz},
  title =	{{Heuristics for Exact 1-Planarity Testing}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.4},
  URN =		{urn:nbn:de:0030-drops-249909},
  doi =		{10.4230/LIPIcs.GD.2025.4},
  annote =	{Keywords: 1-Planarity, Experiments, Backtracking}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.32
Planar Stories of Graph Drawings: Algorithms and Experiments

Authors: Carla Binucci, Sabine Cornelsen, Walter Didimo, Seok-Hee Hong, Eleni Katsanou, Maurizio Patrignani, Antonios Symvonis, and Samuel Wolf

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
We address the problem of computing a dynamic visualization of a geometric graph G as a sequence of frames. Each frame shows only a portion of the graph but their union covers G entirely. The two main requirements of our dynamic visualization are: (i) guaranteeing drawing stability, so to preserve the user’s mental map; (ii) keeping the visual complexity of each frame low. To satisfy the first requirement, we never change the position of the vertices. Regarding the second requirement, we avoid edge crossings in each frame. More precisely, in the first frame we visualize a suitable subset of non-crossing edges; in each subsequent frame, exactly one new edge enters the visualization and all the edges that cross with it are deleted. We call such a sequence of frames a planar story of G. Our goal is to find a planar story whose minimum number of edges contemporarily displayed is maximized (i.e., a planar story that maximizes the minimum frame size). Besides studying our model from a theoretical point of view, we also design and experimentally compare different algorithms, both exact techniques and heuristics. These algorithms provide an array of alternative trade-offs between efficiency and effectiveness, also depending on the structure of the input graph.
Cite as

Carla Binucci, Sabine Cornelsen, Walter Didimo, Seok-Hee Hong, Eleni Katsanou, Maurizio Patrignani, Antonios Symvonis, and Samuel Wolf. Planar Stories of Graph Drawings: Algorithms and Experiments. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{binucci_et_al:LIPIcs.GD.2025.32,
  author =	{Binucci, Carla and Cornelsen, Sabine and Didimo, Walter and Hong, Seok-Hee and Katsanou, Eleni and Patrignani, Maurizio and Symvonis, Antonios and Wolf, Samuel},
  title =	{{Planar Stories of Graph Drawings: Algorithms and Experiments}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.32},
  URN =		{urn:nbn:de:0030-drops-250182},
  doi =		{10.4230/LIPIcs.GD.2025.32},
  annote =	{Keywords: Graph Drawing, Dynamic Graphs, Graph Stories, Heuristics, ILP}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.15
Linear Layouts Revisited: Stacks, Queues, and Exact Algorithms

Authors: Thomas Depian, Simon D. Fink, Robert Ganian, and Vaishali Surianarayanan

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
In spite of the extensive study of stack and queue layouts, many fundamental questions remain open concerning the complexity-theoretic frontiers for computing stack and queue layouts. A stack (resp. queue) layout places vertices along a line and assigns edges to pages so that no two edges on the same page are crossing (resp. nested). We provide three new algorithms which together substantially expand our understanding of these problems: 1) A fixed-parameter algorithm for computing minimum-page stack and queue layouts w.r.t. the vertex integrity of an n-vertex graph G. This result is motivated by an open question in the literature and generalizes the previous algorithms parameterizing by the vertex cover number of G. The proof relies on a newly developed Ramsey pruning technique. Vertex integrity intuitively measures the vertex deletion distance to a subgraph with only small connected components. 2) An n^𝒪(q 𝓁) algorithm for computing 𝓁-page stack and queue layouts of page width at most q. This is the first algorithm avoiding a double-exponential dependency on the parameters. The page width of a layout measures the maximum number of edges one needs to cross on any page to reach the outer face. 3) A 2^𝒪(n) algorithm for computing 1-page queue layouts. This improves upon the previously fastest n^𝒪(n) algorithm and can be seen as a counterpart to the recent subexponential algorithm for computing 2-page stack layouts [ICALP'24], but relies on an entirely different technique.
Cite as

Thomas Depian, Simon D. Fink, Robert Ganian, and Vaishali Surianarayanan. Linear Layouts Revisited: Stacks, Queues, and Exact Algorithms. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{depian_et_al:LIPIcs.ESA.2025.15,
  author =	{Depian, Thomas and Fink, Simon D. and Ganian, Robert and Surianarayanan, Vaishali},
  title =	{{Linear Layouts Revisited: Stacks, Queues, and Exact Algorithms}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.15},
  URN =		{urn:nbn:de:0030-drops-244835},
  doi =		{10.4230/LIPIcs.ESA.2025.15},
  annote =	{Keywords: stack layouts, queue layouts, parameterized algorithms, vertex integrity, Ramsey theory}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.5
On Planar Straight-Line Dominance Drawings

Authors: Patrizio Angelini, Michael A. Bekos, Giuseppe Di Battista, Fabrizio Frati, Luca Grilli, and Giacomo Ortali

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We study the following question, which has been considered since the 90’s: Does every st-planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed y-coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of st-planar graphs that always admit a planar straight-line dominance drawing. These include st-planar 3-trees in which every stacking operation introduces two edges incoming into the new vertex, st-planar graphs in which every vertex is adjacent to the sink, and st-planar graphs in which no face has the left boundary that is a single edge.
Cite as

Patrizio Angelini, Michael A. Bekos, Giuseppe Di Battista, Fabrizio Frati, Luca Grilli, and Giacomo Ortali. On Planar Straight-Line Dominance Drawings. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{angelini_et_al:LIPIcs.WADS.2025.5,
  author =	{Angelini, Patrizio and Bekos, Michael A. and Di Battista, Giuseppe and Frati, Fabrizio and Grilli, Luca and Ortali, Giacomo},
  title =	{{On Planar Straight-Line Dominance Drawings}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.5},
  URN =		{urn:nbn:de:0030-drops-242361},
  doi =		{10.4230/LIPIcs.WADS.2025.5},
  annote =	{Keywords: st-graphs, dominance drawings, planar straight-line drawings, upward planarity}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.14
Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

Authors: Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista, Michael T. Goodrich, and Martin Nöllenburg

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the {dependency digraph} G_𝒫(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V(G_𝒫(I))| √δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√{nm}) time, which improves the best known classical upper bound when m ∈ Ω(n^{1.4}).
Cite as

Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista, Michael T. Goodrich, and Martin Nöllenburg. Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{caroppo_et_al:LIPIcs.WADS.2025.14,
  author =	{Caroppo, Susanna and Da Lozzo, Giordano and Di Battista, Giuseppe and Goodrich, Michael T. and N\"{o}llenburg, Martin},
  title =	{{Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{14:1--14:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.14},
  URN =		{urn:nbn:de:0030-drops-242454},
  doi =		{10.4230/LIPIcs.WADS.2025.14},
  annote =	{Keywords: Dynamic Programming, Quantum Algorithms, Quantum Random Access Memory}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.27
Cops and Robbers for Graphs on Surfaces with Crossings

Authors: Prosenjit Bose, Pat Morin, and Karthik Murali

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber’s turn. In the first round, the cops place themselves on a subset of vertices, after which the robber chooses a vertex to place himself. From the next round onwards, in the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. The cops win if they can capture the robber within a finite number of rounds; else the robber wins. A natural question in this game concerns the cop-number of a graph - the minimum number of cops needed to capture a robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, it was shown recently that the class of 1-planar graphs - graphs that can be drawn on the plane with at most one crossing per edge - does not have bounded cop-number. This paper initiates an investigation into how the distance between crossing pairs of edges influences a graph’s cop number. In particular, we look at Distance d Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance d of the robber. Let c_d(G) denote the minimum number of cops required in the graph G to capture a robber within distance d. We look at various classes of graphs, such as 1-plane graphs, k-plane graphs (graphs where each edge is crossed at most k times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then c_d(G) is bounded, for small values of d. For example, we show that if a graph G admits a drawing in which every pair of crossing edges is contained in a path of length at most 3, then c₄(G) ≤ 21. And if the drawing permits a stronger assumption that the endpoints of every crossing induce the complete graph K₄, then c₃(G) ≤ 9. The tools and techniques that we develop in this paper are sufficiently general, enabling us to examine graphs drawn not only on the sphere but also on orientable and non-orientable surfaces.
Cite as

Prosenjit Bose, Pat Morin, and Karthik Murali. Cops and Robbers for Graphs on Surfaces with Crossings. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.MFCS.2025.27,
  author =	{Bose, Prosenjit and Morin, Pat and Murali, Karthik},
  title =	{{Cops and Robbers for Graphs on Surfaces with Crossings}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.27},
  URN =		{urn:nbn:de:0030-drops-241349},
  doi =		{10.4230/LIPIcs.MFCS.2025.27},
  annote =	{Keywords: Cops and Robbers, Crossings, 1-Planar, Surfaces}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.47
Shelling and Sinking Graphs on the Sphere

Authors: Jeff Erickson and Christian Howard

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We describe a promising approach to efficiently morph spherical graphs, extending earlier approaches of Awartani and Henderson [Trans. AMS 1987] and Kobourov and Landis [JGAA 2006]. Specifically, we describe two methods to morph shortest-path triangulations of the sphere by moving their vertices along longitudes into the southern hemisphere; we call a triangulation sinkable if such a morph exists. Our first method generalizes a longitudinal shelling construction of Awartani and Henderson; a triangulation is sinkable if a specific orientation of its dual graph is acyclic. We describe a simple polynomial-time algorithm to find a longitudinally shellable rotation of a given spherical triangulation, if one exists; we also construct a spherical triangulation that has no longitudinally shellable rotation. Our second method is based on a linear-programming characterization of sinkability. By identifying its optimal basis, we show that this linear program can be solved in O(n^{ω/2}) time, where ω is the matrix-multiplication exponent, assuming the underlying linear system is non-singular. Finally, we pose several conjectures and describe experimental results that support them.
Cite as

Jeff Erickson and Christian Howard. Shelling and Sinking Graphs on the Sphere. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{erickson_et_al:LIPIcs.SoCG.2025.47,
  author =	{Erickson, Jeff and Howard, Christian},
  title =	{{Shelling and Sinking Graphs on the Sphere}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{47:1--47:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.47},
  URN =		{urn:nbn:de:0030-drops-231996},
  doi =		{10.4230/LIPIcs.SoCG.2025.47},
  annote =	{Keywords: morphing, planar graphs, spherical graph drawing, longitudinal shelling}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.65
Recognizing 2-Layer and Outer k-Planar Graphs

Authors: Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. In both cases, edges are drawn as straight-line segments. Both variants are NP-hard, but admit FPT-algorithms with respect to the natural parameter. In recent years, in the context of beyond-planar graphs, a local version of the crossing number has also received considerable attention. A graph is k-planar if it admits a drawing with at most k crossings per edge. In contrast to the crossing number, recognizing k-planar graphs is NP-hard even if k = 1 and hence not likely to be FPT with respect to the natural parameter k. In this paper, we consider the two above variants in the local setting. The k-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer k-planar and outer k-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to the natural parameter k. For k = 0, the two classes of graphs are exactly the caterpillars and outerplanar graphs, respectively, which can be recognized in linear time. Two groups of researchers independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked explicitly whether outer 2-planar graphs can be recognized in polynomial time. Our main contribution consists of XP-algorithms for recognizing 2-layer k-planar graphs and outer k-planar graphs, which implies that both recognition problems can be solved in polynomial time for every fixed k. We complement these results by showing that recognizing 2-layer k-planar graphs is XNLP-complete and that recognizing outer k-planar graphs is XNLP-hard. This implies that both problems are W[t]-hard for every t and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer k-planar graphs where the order of the vertices on one layer is specified.
Cite as

Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff. Recognizing 2-Layer and Outer k-Planar Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kobayashi_et_al:LIPIcs.SoCG.2025.65,
  author =	{Kobayashi, Yasuaki and Okada, Yuto and Wolff, Alexander},
  title =	{{Recognizing 2-Layer and Outer k-Planar Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{65:1--65:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.65},
  URN =		{urn:nbn:de:0030-drops-232170},
  doi =		{10.4230/LIPIcs.SoCG.2025.65},
  annote =	{Keywords: 2-layer k-planar graphs, outer k-planar graphs, recognition algorithms, local crossing number, bandwidth, FPT, XNLP, XP, W\lbrackt\rbrack}
}
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