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LIPIcs, Volume 396
34th International Symposium on Graph Drawing and Network Visualization (GD 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Graph Drawing and Network Visualization (GD)
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- published at: 2026-11-12
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-452-9
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Complete Volume
LIPIcs, Volume 396, GD 2026, Complete Volume
Abstract
LIPIcs, Volume 396, GD 2026, Complete Volume
Cite as
34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 1-716, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{loffler_et_al:LIPIcs.GD.2026,
title = {{LIPIcs, Volume 396, GD 2026, Complete Volume}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {1--716},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026},
URN = {urn:nbn:de:0030-drops-281037},
doi = {10.4230/LIPIcs.GD.2026},
annote = {Keywords: LIPIcs, Volume 396, GD 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{loffler_et_al:LIPIcs.GD.2026.0,
author = {L\"{o}ffler, Maarten and Miksch, Silvia},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {0:i--0:xviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.0},
URN = {urn:nbn:de:0030-drops-281027},
doi = {10.4230/LIPIcs.GD.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Graph-Based Signatures in Topological Data Analysis (Invited Talk)
Abstract
Topological data analysis (TDA) is a growing field that extracts topological summaries from data by leveraging ideas from algebraic topology as well as tools from computational geometry. Rather than measuring distances precisely, TDA asks coarser questions: how many connected components does a dataset have? How many loops? How does shape change across scales? The answers are encoded in topological signatures, which are compact descriptors that are stable under noise and useful in many settings. While persistent homology is perhaps the most famous example, other particularly rich sets of descriptors are graph-based. For example, given a scalar function on a topological space, the Reeb graph captures how the level sets of that function connect and merge, yielding a graph whose structure reflects the topology of the underlying domain. The mapper graph is a practical, combinatorial approximation of the Reeb graph well-suited to point cloud data. Merge trees record the birth and death of connected components as a threshold sweeps through function values, forming a rooted tree that encodes hierarchical structure. Vineyards track how these persistence diagrams evolve as the underlying function changes continuously over time, yielding a dynamic, film-strip view of topological change. These objects sit naturally at the intersection of graph theory and topology, and raise compelling algorithmic questions. This talk will introduce several of these graph-based signatures, with attention to their combinatorial structure and the computational complexity of comparing and visualizing them, with the goal of giving graph drawing researchers both a foothold in TDA and a sense of where their expertise might open new doors.
Biography: Dr. Erin Wolf Chambers is the Schnyder Mission Family Collegiate Professor in the Department of Computer Science and Engineering at the University of Notre Dame, with a concurrent appointment in the Department of Applied and Computational Mathematics and Statistics. Her research focus is on computational topology and geometry, with a more general interest in combinatorics and combinatorial algorithms. She serves on the CGWeek Steering Committees and on the SafeTOC organizing committee, as well as being an editor for the Journal of Applied and Computational Topology. Prior to joining Notre Dame, she was a professor at Saint Louis University from 2008-2024, where she served as Department Chair from 2022-2024. She received her PhD in Computer Science in 2008 and her MS in Mathematics in 2006 from the University of Illinois at Urbana-Champaign, and was a Visiting Research Professor at Saarland University in summer 2011.
Cite as
Erin Chambers. Graph-Based Signatures in Topological Data Analysis (Invited Talk). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chambers:LIPIcs.GD.2026.1,
author = {Chambers, Erin},
title = {{Graph-Based Signatures in Topological Data Analysis}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.1},
URN = {urn:nbn:de:0030-drops-278963},
doi = {10.4230/LIPIcs.GD.2026.1},
annote = {Keywords: graphs, signatures, topological data analysis}
}
Document
Invited Talk
31 Years of Graph Drawing: Analyzing the Oddballs (Invited Talk)
Abstract
In reflecting on 13 graph drawing projects done across 31 years, I realized I have a fondness for the oddballs - the strangest ones. Analyzing these projects through the lens of my nested model of visualization design sheds some light on why. The nested model splits up visualization design decisions into four levels: domain, abstraction, idiom, and algorithm. There’s a lot of solid work to be done at the idiom and algorithm levels, and that’s where I started with graph drawing. However, it’s at the abstraction level, where the specifics of the data and tasks drive the design, that things get really interesting!
Biography: Tamara Munzner is a Professor at the University of British Columbia Department of Computer Science, and holds a 2000 PhD from Stanford. She has been papers chair for IEEE VIS, IEEE InfoVis, and EuroVis. She chaired the InfoVis Steering Committee, the VIS Executive Committee, and the reVISe Committee. Her book Visualization Analysis and Design is widely used world-wide, and she is the co-editor of the A K Peters Visualization book series at CRC/Routledge. She has published over 100 papers, received the IEEE VGTC Visualization Technical Achievement Award and multiple Test of Time Awards from InfoVis, is a member of the VIS Academy, and is a Fellow of the IEEE and the Royal Society of Canada.
Cite as
Tamara Munzner. 31 Years of Graph Drawing: Analyzing the Oddballs (Invited Talk). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{munzner:LIPIcs.GD.2026.2,
author = {Munzner, Tamara},
title = {{31 Years of Graph Drawing: Analyzing the Oddballs}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.2},
URN = {urn:nbn:de:0030-drops-278971},
doi = {10.4230/LIPIcs.GD.2026.2},
annote = {Keywords: graph drawing, nested model of visualization design}
}
Document
How to Draw a Planar Graph: An Experimental Evaluation
Abstract
Planar graphs are central to graph drawing, with extensive results on planar layouts and related structures. Every planar graph admits a planar straight-line drawing, and algorithms can guarantee additional geometric or combinatorial properties. However, it is unclear which algorithms work best in practice. Even for small graphs with near-perfect manual drawings, standard algorithms might produce poor spacing, distorted faces, or small angles.
We present an experimental evaluation of planar graph drawing algorithms on a large benchmark collection of small and medium-sized planar graphs (10-400 vertices). The study compares established algorithms from the graph drawing literature, practical force-directed and pressure-based heuristics, and new optimization-based methods that directly improve visual properties such as edge-length uniformity, face-area balance, and angular resolution. The results show that no evaluated algorithm is best across all aesthetic criteria, and optimizing one visual property often worsens another. Directly optimizing visual criteria improves targeted scores, and score-guided combination of several methods gives the best aggregate results, but no simple algorithm emerges as a clear universal default. Designing a simple, robust algorithm that performs well across graph families and aesthetic criteria therefore remains an open practical problem.
Cite as
Sergey Pupyrev. How to Draw a Planar Graph: An Experimental Evaluation. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pupyrev:LIPIcs.GD.2026.3,
author = {Pupyrev, Sergey},
title = {{How to Draw a Planar Graph: An Experimental Evaluation}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {3:1--3:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.3},
URN = {urn:nbn:de:0030-drops-278985},
doi = {10.4230/LIPIcs.GD.2026.3},
annote = {Keywords: Planar graph drawing, drawing aesthetics, experimental evaluation}
}
Document
Extending Biconnected Straight-Line Planar Drawings
Abstract
The Partial Drawing Extensibility problem, for short PDE, takes as input a triple ⟨ G,H,Γ_H⟩, where G is a planar graph, H is a subgraph of G, and Γ_H is a straight-line planar drawing of H, and asks whether Γ_H can be extended to a straight-line planar drawing of G. Patrignani [Int. J. Found. Comput. Sci. (2006)] proved that the PDE problem is NP-hard, exploiting instances in which H is highly disconnected. In this paper, we study the PDE problem under the requirement that the initial partial drawing Γ_H is biconnected. We show that PDE remains NP-hard even for instances in which H is a biconnected graph with faces of bounded size, G is subcubic, and the part of G that is not in H consists of length-2 paths. The complexity of PDE remains however open when H is connected (or even biconnected) if G has a fixed embedding. In this setting both a polynomial-time algorithm or an NP-hardness proof seem to be elusive targets. As a step towards tackling this problem, we study instances of PDE in which H is biconnected, G has a fixed embedding, and the rest of the graph consists of p length-2 paths, and present an O(p² n)-time algorithm, a result in sharp contrast with the NP-hardness of the variable embedding setting. Moreover, with an approach based on the Existential Theory of the Reals, we show that, if H is biconnected, the problem is FPT parameterized by the vertex cover number of G, both in a fixed and in a variable embedding setting.
Cite as
Giordano Andreola, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani. Extending Biconnected Straight-Line Planar Drawings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{andreola_et_al:LIPIcs.GD.2026.4,
author = {Andreola, Giordano and Caroppo, Susanna and Da Lozzo, Giordano and D'Elia, Marco and Di Battista, Giuseppe and Frati, Fabrizio and Grosso, Fabrizio and Patrignani, Maurizio},
title = {{Extending Biconnected Straight-Line Planar Drawings}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {4:1--4:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.4},
URN = {urn:nbn:de:0030-drops-278995},
doi = {10.4230/LIPIcs.GD.2026.4},
annote = {Keywords: Planar Graphs, Straight-line Drawings, Extension Problems, FPT, Fixed Embedding}
}
Document
Beyond Degree Four: Near-Orthogonal Planar Drawings
Abstract
Orthogonal planar drawings constitute a classical and mainstream research topic in graph drawing due to their clarity and wide applicability. In an orthogonal planar drawing of a graph, each face is represented as an orthogonal polygon, that is, a polygon whose edges are either horizontal or vertical. Yet a planar graph admits such a representation if and only if its maximum degree is at most four.
In this paper, we consider planar polyline drawings of graphs with unrestricted maximum degree. We focus on drawings that are "close to orthogonal", where closeness is measured by the number of faces that are not orthogonal polygons. We show that, even when the input graph is triconnected and thus has a unique planar embedding, the problem of testing whether there exists a planar polyline drawing with at most h non-orthogonal faces is NP-complete. Motivated by this computational hardness, we study parameterized and approximation algorithms. In the fixed-embedding setting, we prove that the problem admits linear-time FPT algorithms parameterized by (i) the outerplanarity index and (ii) the natural parameter h. In addition, we provide an FPT algorithm parameterized by the treewidth and a polynomial-time approximation scheme. In the variable-embedding setting, we give an FPT algorithm parameterized by treewidth for biconnected graphs.
Cite as
Patrizio Angelini, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, and Ignaz Rutter. Beyond Degree Four: Near-Orthogonal Planar Drawings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{angelini_et_al:LIPIcs.GD.2026.5,
author = {Angelini, Patrizio and Cornelsen, Sabine and Da Lozzo, Giordano and Hong, Seok-Hee and Rutter, Ignaz},
title = {{Beyond Degree Four: Near-Orthogonal Planar Drawings}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {5:1--5:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.5},
URN = {urn:nbn:de:0030-drops-279003},
doi = {10.4230/LIPIcs.GD.2026.5},
annote = {Keywords: planar orthogonal drawings, unbounded degree, FPT, PTAS, treewidth}
}
Document
A Fixed-Parameter Algorithm for Extending Upward Planar Drawings
Abstract
An upward planar drawing of a directed acyclic graph is a planar drawing where every edge is pointed upward from its tail to head. Upward planar drawings are among the most natural drawing styles of directed graphs and have been researched in a variety of different settings, recently including that of drawing extension. In the drawing extension setting, one asks: given a graph G and a (typically connected) subgraph H of G with a drawing Γ(H), can we complete Γ(H) to a drawing of G?
Drawing extension problems have been studied for numerous drawing styles; the vast majority of these are NP-hard and a typical approach aimed at circumventing their intractability is to design parameterized algorithms where the parameter measures "how much" of G is still missing from the pre-drawn graph H. Most algorithms obtained within this framework require only a small number of edges to be missing from H in order to remain efficient. In this article, we present a fixed-parameter algorithm for extending upward planar drawings which overcomes this drawback by using the vertex+edge deletion distance as the parameter, thus achieving tractability even for instances with many missing edges. A key ingredient towards our result is a novel characterization of "canonical" sets of missing edges which cross a horizontal line segment in the drawing.
Cite as
Vera Chekan, Robert Ganian, and Viktoriia Korchemna. A Fixed-Parameter Algorithm for Extending Upward Planar Drawings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chekan_et_al:LIPIcs.GD.2026.6,
author = {Chekan, Vera and Ganian, Robert and Korchemna, Viktoriia},
title = {{A Fixed-Parameter Algorithm for Extending Upward Planar Drawings}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {6:1--6:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.6},
URN = {urn:nbn:de:0030-drops-279014},
doi = {10.4230/LIPIcs.GD.2026.6},
annote = {Keywords: upward planar drawings, extension problems, parameterized complexity}
}
Document
Minimum-Width Drawing of Trees with Sized Vertices
Abstract
Trees arise in many applications and computing nice tree layouts is a classical problem in information visualization. In many practical settings, vertices need to be represented as rectangles with a given width and height rather than as points. When an order over the children of each vertex is given, polynomial-time algorithms are known that produce drawings adhering to various drawing conventions. However, in many applications, the order of children carries no semantic meaning, and choosing it well can significantly reduce the drawing’s width. In this paper, we study the problem Min-Width Tree Drawing with Reordering (MWD): given a rooted tree whose vertices have prescribed widths, find a sibling order at each internal vertex that minimizes the width of the resulting layered drawing. We show that the problem is NP-complete, even on binary trees with unit-width vertices. We present a mixed integer linear program that solves MWD exactly on moderately sized instances, and a heuristic that is fast and delivers good results in practice. We evaluate both approaches against a baseline on synthetic and real-world datasets, where reordering reduces drawing width by a median of ≈20% and by up to ≈55% on individual instances. The heuristic computes its layouts in under a second and, when the MILP proves optimality, it stays within 25% of the optimal width in three-quarters of all instances.
Cite as
Markus Wallinger, Oscar Navarro, and Stephen G. Kobourov. Minimum-Width Drawing of Trees with Sized Vertices. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wallinger_et_al:LIPIcs.GD.2026.7,
author = {Wallinger, Markus and Navarro, Oscar and Kobourov, Stephen G.},
title = {{Minimum-Width Drawing of Trees with Sized Vertices}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {7:1--7:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.7},
URN = {urn:nbn:de:0030-drops-279028},
doi = {10.4230/LIPIcs.GD.2026.7},
annote = {Keywords: Tree drawing, sized vertices, minimum width, integer linear programming, heuristics}
}
Document
Recognizing Penny and Marble Graphs Is Hard for Existential Theory of the Reals
Abstract
We show that the recognition problem for penny graphs (contact graphs of unit disks in the plane) is ∃ℝ-complete, that is, computationally as hard as the existential theory of the reals, even if a combinatorial plane embedding of the graph is given. The exact complexity of the penny graph recognition problem has been a long-standing open problem.
We lift the penny graph result to three dimensions and show that the recognition problem for marble graphs (contact graphs of unit balls in three dimensions) is ∃ℝ-complete.
Finally, we show that rigidity of penny graphs is ∀ℝ-complete and look at grid embeddings of penny graphs that are trees.
Cite as
Anna Lubiw and Marcus Schaefer. Recognizing Penny and Marble Graphs Is Hard for Existential Theory of the Reals. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lubiw_et_al:LIPIcs.GD.2026.8,
author = {Lubiw, Anna and Schaefer, Marcus},
title = {{Recognizing Penny and Marble Graphs Is Hard for Existential Theory of the Reals}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {8:1--8:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.8},
URN = {urn:nbn:de:0030-drops-279030},
doi = {10.4230/LIPIcs.GD.2026.8},
annote = {Keywords: penny graphs, contact graphs of unit disks, marble graphs, contact graphs of balls, \exists\mathbb{R}, existential theory of the reals, computational complexity, rigidity}
}
Document
How Close Is a Tree to a Euclidean Minimum Spanning Tree?
Abstract
Let Γ be a straight-line crossing-free drawing of a tree T. A bad pair in Γ is a pair of non-adjacent vertices of T whose Euclidean distance in Γ is smaller than the length of the longest edge in the path connecting them in Γ. When Γ has no bad pairs, Γ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For n-vertex trees with maximum vertex degree Δ, we prove the Δ²nlog n upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.
Cite as
Todor Antić, Jiří Fiala, Jelena Glišić, Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Jan Kratochvíl, Giuseppe Liotta, Morteza Saghafian, Maria Saumell, Krisztina Szilágyi, and Pavel Valtr. How Close Is a Tree to a Euclidean Minimum Spanning Tree?. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{antic_et_al:LIPIcs.GD.2026.9,
author = {Anti\'{c}, Todor and Fiala, Ji\v{r}{\'\i} and Gli\v{s}i\'{c}, Jelena and Gutowski, Grzegorz and Junosza-Szaniawski, Konstanty and Kratochv{\'\i}l, Jan and Liotta, Giuseppe and Saghafian, Morteza and Saumell, Maria and Szil\'{a}gyi, Krisztina and Valtr, Pavel},
title = {{How Close Is a Tree to a Euclidean Minimum Spanning Tree?}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {9:1--9:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.9},
URN = {urn:nbn:de:0030-drops-279045},
doi = {10.4230/LIPIcs.GD.2026.9},
annote = {Keywords: Straight-line Drawing of Tree, Euclidean Minimum Spanning Tree, EMST-Drawability, Caterpillar, Graph Drawing Algorithm}
}
Document
Paged Geophylogenies: A Coloring Approach to External Labeling with Tree Constraints
Abstract
Geophylogenies are a common type of diagram for visualizing the evolutionary history of species in a geographic context. As a drawing problem, these diagrams are commonly modeled as a rooted binary "phylogenetic" tree T where every leaf is associated with a point feature ("site") in a rectangular map range. The tree is drawn downward planar, its leaves are placed at equidistant positions on the upper boundary of the map, and each leaf is connected to its corresponding site by a straight-line leader. Prior work focuses on minimizing the number of leader crossings, or avoiding leaders altogether. In this paper, we explore paged geophylogenies, where the leaders can be partitioned into multiple pages where only crossings within a page are counted. For the general case, where each page can contain an arbitrary subset of the leaders, we provide an integer linear programming (ILP) formulation for minimizing the number of pages, and a polynomial-time algorithm for a special case that is equivalent to a one-sided tanglegram. We argue that, from a visualization perspective, instead each page must contain only leaders for a single subtree, and provide an 𝒪(n⁷) time algorithm for minimizing pages in this setting - which also involves improving the fastest known algorithm for testing the existence of a crossing-free labeling. Counter to what our worst case bound suggests, our implementation can handle instances with hundreds of sites within a second, as we show in our experimental evaluation, which further investigates the trade-offs between leader crossings and the number of pages.
Cite as
Thomas Depian, Thomas C. van Dijk, and Martin Nöllenburg. Paged Geophylogenies: A Coloring Approach to External Labeling with Tree Constraints. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{depian_et_al:LIPIcs.GD.2026.10,
author = {Depian, Thomas and van Dijk, Thomas C. and N\"{o}llenburg, Martin},
title = {{Paged Geophylogenies: A Coloring Approach to External Labeling with Tree Constraints}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {10:1--10:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.10},
URN = {urn:nbn:de:0030-drops-279052},
doi = {10.4230/LIPIcs.GD.2026.10},
annote = {Keywords: boundary labeling, phylogenetic trees, multi-page layout}
}
Document
Navigating Posets with Few Maps
Abstract
We study two new parameters for finite posets motivated by the problem of efficiently determining the set of successors of a given element. A plane map of a poset P = (X, ≤) is an injective mapping of X into the Cartesian plane ℝ². Given two different points a and b in the plane, we say that b dominates a if a < b coordinatewise. We say that an element x of P is tight in a plane map μ if the following holds: x < y in P if and only if μ(y) dominates μ(x). Note that, by definition, every 2-dimensional poset admits a map such that every element of the poset is tight. For any poset P, we define the mapability of P, dmap(P), to be the maximum number of elements that are tight in a single map, and we define the atlas thickness of P, at(P), to be the size of the smallest collection of maps such that every element is tight in at least one map of the collection.
We relate these parameters to the classical notions of dimension and width: for every poset P, we show that dim(P) ≤ 2at(P) ≤ width(P)+1. On the other hand, there exists a sequence of posets (P_n)_{n ≥ 1} such that the atlas thickness of P_n is doubly exponential in the dimension of P_n.
On the computational side, we prove that it is NP-complete, for a given poset P, to compute the mapability of P and to decide whether at(P) ≤ 2. In contrast to the latter, we show that computing the mapability of a poset is fixed-parameter tractable with respect to the natural parameter.
Cite as
Stefan Felsner, Jędrzej Hodor, Giacomo Ortali, and Alexander Wolff. Navigating Posets with Few Maps. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 11:1-11:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{felsner_et_al:LIPIcs.GD.2026.11,
author = {Felsner, Stefan and Hodor, J\k{e}drzej and Ortali, Giacomo and Wolff, Alexander},
title = {{Navigating Posets with Few Maps}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {11:1--11:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.11},
URN = {urn:nbn:de:0030-drops-279064},
doi = {10.4230/LIPIcs.GD.2026.11},
annote = {Keywords: Poset, dimension, width, atlas thickness, mapability, NP-hard, fixed-parameter tractable}
}
Document
Node Labeling in Line Diagrams of Ordered Sets
Abstract
We propose a flexible, two-phase algorithm for labeling line diagrams of ordered sets, in which the nodes of direct neighbors in the order relation are connected by a straight, upward-pointing line. In contrast to the labeling of diagrams of arbitrary graphs, we benefit from the fact that all edges in line diagrams of ordered sets are more or less vertical. In this paper, we study the placement of all labels such that they do not intersect with any nodes, lines, or other labels while minimizing the distances between the nodes and their labels.
Our approach starts by filtering the fixed-position model using line diagram-specific readability criteria. For labels that cannot be placed adjacent to their node (overflow labels), we exploit the free space in the graph’s interior or the infinite space surrounding the drawing and link the labels with their respective node by straight binding lines that should not cross other nodes or labels if possible. To balance quality and runtime, we derive an initial placement of the overflow labels using a cost function over a sparse grid of candidates, followed by a force-based refinement step to fine-tune the layout.
Furthermore, we demonstrate the flexibility of this approach by applying it to the visual constraints of line diagrams in the field of Formal Concept Analysis (FCA), where certain labels have to be placed above their node and others below. This special version of the algorithm shows that a pre-filtering in the first phase and minimal adjustments for the cost function and force-based model are sufficient to handle the dual labeling requirements of concept lattices.
Cite as
Marcel Nöhre and Gerd Stumme. Node Labeling in Line Diagrams of Ordered Sets. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nohre_et_al:LIPIcs.GD.2026.12,
author = {N\"{o}hre, Marcel and Stumme, Gerd},
title = {{Node Labeling in Line Diagrams of Ordered Sets}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.12},
URN = {urn:nbn:de:0030-drops-279078},
doi = {10.4230/LIPIcs.GD.2026.12},
annote = {Keywords: Node Labeling, Line Diagrams, Formal Concept Analysis}
}
Document
Stack and Queue Layouts with Defects
Abstract
Linear layouts of graphs - particularly stack and queue layouts - are well-established types of representations in graph drawing, thanks to their connection with numerous theoretical and practical problems. In such layouts, all vertices are linearly ordered and the edges are partitioned into sets that avoid specific forbidden configurations: in a stack layout no two independent edges within the same set cross, whereas in a queue layout no two independent edges within the same set are nested. A central problem in this context is to determine, for a given graph G, its stack number or queue number, that is, the minimum number of sets into which the edges can be partitioned so that a corresponding stack or queue layout of G exists. In this work, we introduce a relaxation of stack and queue layouts, which allows some forbidden patterns for the edges in the same set. Namely, for a given integer k > 0, a k-defective stack layout (resp. a k-defective queue layout) allows an edge to be in a crossing (resp. nesting) relationship with at most k edges within the same set. Our motivation is to extend the classes of graphs that admit linear layouts using a limited number of edge-partition sets, at the cost of allowing some defects. We study defective linear layouts both from a combinatorial and from an algorithmic perspective, providing an array of results across different graph classes and parameters.
Cite as
Michael A. Bekos, Carla Binucci, Emilio Di Giacomo, Walter Didimo, Luca Grilli, Maria Eleni Pavlidi, Alessandra Tappini, and Alexandra Weinberger. Stack and Queue Layouts with Defects. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bekos_et_al:LIPIcs.GD.2026.13,
author = {Bekos, Michael A. and Binucci, Carla and Di Giacomo, Emilio and Didimo, Walter and Grilli, Luca and Pavlidi, Maria Eleni and Tappini, Alessandra and Weinberger, Alexandra},
title = {{Stack and Queue Layouts with Defects}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {13:1--13:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.13},
URN = {urn:nbn:de:0030-drops-279086},
doi = {10.4230/LIPIcs.GD.2026.13},
annote = {Keywords: Linear layouts with defects, stack layouts, queue layouts}
}
Document
Monotone Clustered Level Planarity
Abstract
We consider the combination of the two constrained planarity problems Level- and Clustered Planarity. Traditionally, level-planar drawings with convex clusters have been studied in this setting. Fink et al. [Simon D. Fink et al., 2024] recently introduced a different way of combining level- and clustered planarity by mimicking a classic characterization of clustered planarity in the level-planar setting: The problem (y-)monotone Clustered Level Planarity (mCLP) seeks a level-planar drawing in which it is possible to augment each cluster with edges that do not cross cluster boundaries so that it becomes connected while maintaining level-planarity. This is in line with previous research on clustered planarity that poses certain requirements on the augmentation edges that make each cluster connected, e.g., that they form a path. Fink et al. [Simon D. Fink et al., 2024] showed that mCLP is NP-complete even for biconnected single-source graphs and instances with a constant number of levels and clusters.
We further classify the parameterized complexity of the mCLP problem by, on the one hand, showing hardness even for instances that consist of a forest with trees of bounded size, no isolated vertices, and a small constant number of either clusters or levels. This excludes fixed-parameter tractability for almost all graph-structural parameters, except for vertex cover, even in conjunction with the number of clusters. We complement this by showing fixed-parameter tractability when parameterizing by the vertex cover number and the number of clusters. A major obstacle that we overcome is the fact that mCLP is non-hereditary, i.e., subinstances of yes-instances may be no-instances and vice versa, which makes it challenging to apply usual reduction techniques.
Cite as
Simon D. Fink, Matthias Pfretzschner, Ignaz Rutter, and Marie Diana Sieper. Monotone Clustered Level Planarity. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fink_et_al:LIPIcs.GD.2026.14,
author = {Fink, Simon D. and Pfretzschner, Matthias and Rutter, Ignaz and Sieper, Marie Diana},
title = {{Monotone Clustered Level Planarity}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.14},
URN = {urn:nbn:de:0030-drops-279098},
doi = {10.4230/LIPIcs.GD.2026.14},
annote = {Keywords: Level Planarity, Clustered Planarity, Cluster Augmentation, Parameterized Complexity}
}
Document
On 2-Layer k-Matching-Planar Graphs
Abstract
A graph is k-matching-planar if it admits a drawing in the plane such that, for every edge e, the edges crossing e contain no matching of size greater than k. The class of k-matching-planar graphs generalizes other beyond-planar graph classes, such as k-planar and fan-planar graphs. In a 2-layer drawing of a bipartite graph, the vertices of the two bipartition classes are placed on two parallel horizontal lines and edges are drawn as straight-line segments between them.
We prove that every graph with a 2-layer k-matching-planar drawing has pathwidth at most 2k+1. Moreover, for every k ≥ 0, we construct a graph with a 2-layer k-matching-planar drawing whose pathwidth is 3⌊k/2⌋ + 1. On the algorithmic side, we consider the one-sided recognition problem where a fixed embedding of the vertices on one side is given. We show that this problem is NP-hard. On the other hand, we prove that the problem is fixed-parameter tractable with respect to k. Finally, we prove that the two-sided variant cannot be approximated within any constant factor in polynomial time unless P = NP.
Cite as
Saeed Odak, Jonathan Rollin, and Torben Scheele. On 2-Layer k-Matching-Planar Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{odak_et_al:LIPIcs.GD.2026.15,
author = {Odak, Saeed and Rollin, Jonathan and Scheele, Torben},
title = {{On 2-Layer k-Matching-Planar Graphs}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {15:1--15:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.15},
URN = {urn:nbn:de:0030-drops-279108},
doi = {10.4230/LIPIcs.GD.2026.15},
annote = {Keywords: pathwidth, k-matching-planar graph, 2-layer drawings, NP-hardness, k-planar graph, one-sided graph drawing}
}
Document
Two-Layer Drawings with a Tree on Top: Vertex Splits and Fixed-Parameter Algorithms
Abstract
Two-layer drawings of bipartite graphs place the vertices of each part on one of two parallel lines and draw the edges as straight-line links. Traditionally, the optimization goal is to find vertex permutations on one or both layers that minimize the induced number of edge crossings. This problem is NP-hard, and crossing-minimal solutions may still contain many crossings. Recently, there has been growing interest in an orthogonal optimization goal, namely removing all crossings by vertex splitting, i.e., replacing original vertices by two or more copies and distributing the adjacencies among them. In this paper, we study a natural extension of the two-layer vertex splitting problem in which the vertex order on one layer is constrained by a given auxiliary tree T, motivated by applications such as the visualization of anatomical hierarchies in the Human Reference Atlas. We investigate the parameterized complexity of this problem and obtain two main contributions: (1) a fixed-parameter algorithm with respect to the number k of splits, and (2) an ETH-tight single-exponential fixed-parameter algorithm with respect to the maximum degree of T. Moreover, we build on the latter result to obtain an ETH-tight single-exponential algorithm for the classical unconstrained version of the problem, improving upon the previous 𝒪^*(2^{k⋅log k}) algorithms. Finally, we also implement our algorithm and show that it performs well in practice.
Cite as
Alexander Firbas, Robert Ganian, Sylvain Meunier, and Martin Nöllenburg. Two-Layer Drawings with a Tree on Top: Vertex Splits and Fixed-Parameter Algorithms. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{firbas_et_al:LIPIcs.GD.2026.16,
author = {Firbas, Alexander and Ganian, Robert and Meunier, Sylvain and N\"{o}llenburg, Martin},
title = {{Two-Layer Drawings with a Tree on Top: Vertex Splits and Fixed-Parameter Algorithms}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {16:1--16:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.16},
URN = {urn:nbn:de:0030-drops-279117},
doi = {10.4230/LIPIcs.GD.2026.16},
annote = {Keywords: two-layer graph drawing, vertex splitting, fixed-parameter algorithms}
}
Document
2-Layer Fan-Planarity in Polynomial Time
Abstract
In this paper, we give a polynomial-time algorithm for deciding whether a bipartite graph admits a 2-layer fan-planar drawing. This resolves an open problem posed in several papers since 2015.
Cite as
Yasuaki Kobayashi and Yuto Okada. 2-Layer Fan-Planarity in Polynomial Time. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 17:1-17:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kobayashi_et_al:LIPIcs.GD.2026.17,
author = {Kobayashi, Yasuaki and Okada, Yuto},
title = {{2-Layer Fan-Planarity in Polynomial Time}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {17:1--17:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.17},
URN = {urn:nbn:de:0030-drops-279123},
doi = {10.4230/LIPIcs.GD.2026.17},
annote = {Keywords: 2-layer drawing, fan-planar drawing, bipartite graphs}
}
Document
On the Complexity of Extending Storylines
Abstract
Storyline layouts visualize temporal interactions by drawing each character as an x-monotone curve and enforcing that the participants of every meeting form a contiguous vertical group. We study a drawing extension variant in which a layout of a sub-storyline is fixed and has to be extended by inserting k missing characters while adhering to all meeting constraints. We consider two optimization objectives: minimizing the number of additional crossings introduced to complete the storyline, and minimizing the maximum number of crossings incurred by any single character. For both variants, we analyze the parameterized complexity with respect to the natural parameter k representing the size of the missing information, as well as additional structural parameters such as the number of characters σ per time instant and the number of meetings μ involving missing characters. We contribute a broad collection of results, most of them tight, drawing a nearly complete picture of the complexity landscape of these extension problems.
Cite as
Alexander Dobler, Siddharth Gupta, Philipp Kindermann, Fabrizio Montecchiani, and Martin Nöllenburg. On the Complexity of Extending Storylines. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 18:1-18:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dobler_et_al:LIPIcs.GD.2026.18,
author = {Dobler, Alexander and Gupta, Siddharth and Kindermann, Philipp and Montecchiani, Fabrizio and N\"{o}llenburg, Martin},
title = {{On the Complexity of Extending Storylines}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {18:1--18:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.18},
URN = {urn:nbn:de:0030-drops-279131},
doi = {10.4230/LIPIcs.GD.2026.18},
annote = {Keywords: Storyline visualization, (local) crossing number, parameterized complexity}
}
Document
Clic: Highlighting Clique Motifs in Biofabric Visualizations
Abstract
Dense graphs are hard to read because many edges compete for the same visual space. BioFabric reduces this problem by drawing nodes as horizontal lines and edges as vertical segments, removing edge crossings by design. However, its readability still depends strongly on the ordering of nodes and edges. In this paper, we focus on clique motifs in BioFabric. Since every pair of nodes in a clique is connected, drawing all clique edges can add visual clutter without adding much information once the clique is known. A compact square or rectangular mark can instead indicate the clique as a dense region of the network. This only works, however, when the layout places the clique nodes and edges close enough to form a contiguous region. We model this as an optimization problem and introduce an integer linear programming formulation that computes node orderings designed to make clique motifs compact. We provide a quantitative evaluation to show the scalability and results obtained with the method. Our approach extends recent work on BioFabric layout optimization from staircase motifs to clique motifs, with particular attention to overlapping cliques, where several compact regions compete for the same rows.
Cite as
Sara Di Bartolomeo, Fabrizio Montecchiani, and Andrea Tomassoni. Clic: Highlighting Clique Motifs in Biofabric Visualizations. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dibartolomeo_et_al:LIPIcs.GD.2026.19,
author = {Di Bartolomeo, Sara and Montecchiani, Fabrizio and Tomassoni, Andrea},
title = {{Clic: Highlighting Clique Motifs in Biofabric Visualizations}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {19:1--19:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.19},
URN = {urn:nbn:de:0030-drops-279144},
doi = {10.4230/LIPIcs.GD.2026.19},
annote = {Keywords: Biofabric, Clique Visualization, Clustered Graphs, Integer Linear Programming, Motif Simplification}
}
Document
Tight Leading-Term Bounds for the ℤ₂-Genus of Complete Graphs
Abstract
A drawing of a graph on a surface is independently even if every pair of independent edges in the drawing crosses an even number of times. The Euler ℤ₂-genus eg₀(G) and the orientable ℤ₂-genus g₀(G) are the minimal Euler genus and orientable genus on which the graph G has an independently even drawing.
We prove tight leading-term bounds for eg₀(K_n) and g₀(K_n). For every n ≥ 3, we show that eg₀(K_n) ≥ ⌈((n-3)(n-4) - (2n-5))/6⌉ and g₀(K_n) ≥ ⌈((n-3)(n-4) - (2n-5))/12⌉ .
Together with the classical genus formulas for K_n, these bounds determine the Euler and orientable ℤ₂-genera of complete graphs up to an additive O(n) term. Thus, for complete graphs, passing from embeddings to independently even drawings does not reduce the leading term of the required genus.
The proof adapts a matrix-rank framework previously used for complete bipartite graphs. Starting from an independently even drawing of K_n on a surface of minimum ℤ₂-genus, we planarize the drawing and encode crossing parities in a matrix over ℤ₂. Parity constraints specific to complete graphs allow us to transform this matrix into a structured form whose rank we can lower bound in two different ways, together yielding the stated genus lower bounds.
Cite as
Michal Katrlík. Tight Leading-Term Bounds for the ℤ₂-Genus of Complete Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{katrlik:LIPIcs.GD.2026.20,
author = {Katrl{\'\i}k, Michal},
title = {{Tight Leading-Term Bounds for the \mathbb{Z}₂-Genus of Complete Graphs}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {20:1--20:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.20},
URN = {urn:nbn:de:0030-drops-279159},
doi = {10.4230/LIPIcs.GD.2026.20},
annote = {Keywords: drawing of graphs, genus of a graph, Z2-genus of a graph, Hanani-Tutte theorem}
}
Document
What Induces Plane Structures in Complete Graph Drawings?
Abstract
This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is obeyed: any two adjacent curves do not cross, or any two non-adjacent curves cross at most once. Here, two curves are called adjacent if they share an endpoint. On the other hand, we demonstrate how to draw all curves such that any two adjacent curves cross exactly once, any two non-adjacent curves cross at least once and at most twice, and thus no two curves are disjoint. Furthermore, we analyze the emergence of disjoint curves without these mild assumptions, and characterize the plane structures in complete graph drawings guaranteed by each of the rules above.
Cite as
Alexandra Weinberger and Ji Zeng. What Induces Plane Structures in Complete Graph Drawings?. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{weinberger_et_al:LIPIcs.GD.2026.21,
author = {Weinberger, Alexandra and Zeng, Ji},
title = {{What Induces Plane Structures in Complete Graph Drawings?}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {21:1--21:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.21},
URN = {urn:nbn:de:0030-drops-279160},
doi = {10.4230/LIPIcs.GD.2026.21},
annote = {Keywords: complete graph drawings, disjoint edges, plane structures}
}
Document
Antipodal Pairs and Crossing Numbers
Abstract
This paper makes contributions to the theory of crossing numbers of complete and almost complete graphs. An antipodal pair in a drawing is a pair of vertices such that the stars of the two vertices induce a plane subgraph of the drawing. An optimal drawing of a graph is a drawing with a minimum number of crossings. Hill conjectured that the crossing number of the complete graph is: cr(K_n) = H(n) = 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)2⌋ ⌊(n-3)/2⌋. We establish connections between antipodal pairs and optimal drawings of complete graphs. In particular we show that Hill’s conjecture is true within the class of antipodally shellable drawings.
The graph M_{n}^t is obtained from the complete graph K_n by removing the edges of a matching of size t. For these graphs we conjecture that for large enough n optimal drawings have antipodal pairs. If true this conjecture implies that a conjecture of Mohar about the crossing number of M_{2k}^t and a conjecture of the author about the crossing number of M_{2k+1}^t can be verified by checking "small" cases only.
We also construct a new family of Hill drawings, i.e., a family of drawings of K_n with crossing number equal to H(n). As a consequence of this new construction we obtain that every spherical drawing of K_k is a subdrawing of a Hill drawing of K_n for all n ≥ 2k+1.
Cite as
Stefan Felsner. Antipodal Pairs and Crossing Numbers. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{felsner:LIPIcs.GD.2026.22,
author = {Felsner, Stefan},
title = {{Antipodal Pairs and Crossing Numbers}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.22},
URN = {urn:nbn:de:0030-drops-279173},
doi = {10.4230/LIPIcs.GD.2026.22},
annote = {Keywords: graph drawing, complete graph, crossing number, antipodal pair, Hill conjecture}
}
Document
Weaving (Bio)Fabric: A Benchmark for Scalability Through Progressive Sampling Methods
Abstract
Network data is widely used to represent complex systems across domains, enabling the discovery of structural patterns and relationships. However, as graphs grow in size, visualizing them effectively becomes increasingly challenging. BioFabric offers a compact representation in which nodes are drawn as horizontal lines and edges as vertical lines, eliminating edge crossings by design and supporting the identification of meaningful motifs such as staircases. Despite these advantages, the readability of BioFabric layouts depends critically on the ordering of nodes and edges, making both computation and visualization difficult to scale to high cardinality data. To address these challenges, we explore progressive visualization, a paradigm that allows users to inspect increasingly accurate partial results as data processing continues. Applying progressive visualization to BioFabric introduces a key difficulty: intermediate layouts may contain correct subsets of the data yet remain visually misleading if their ordering diverges significantly from the final layout. We frame this problem as one of ordering convergence to investigate how different sampling strategies affect the evolution of the layout. Specifically, we design a benchmark to evaluate how quickly intermediate visualizations approximate the final BioFabric ordering using measures such as Kendall-τ, and analyze the impact of progressive computation on the emergence of recognizable patterns, such as staircases. Our results provide a first characterization of how graph topology and progressive data chunking strategies influence convergence and layout accuracy. We derive best practices for designing progressive BioFabric systems that become informative before the full computation is complete. Two case studies show the benefits of progressively sampling BioFabric.
Cite as
Alessandro Palma, Sara Di Bartolomeo, and Marco Angelini. Weaving (Bio)Fabric: A Benchmark for Scalability Through Progressive Sampling Methods. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{palma_et_al:LIPIcs.GD.2026.23,
author = {Palma, Alessandro and Di Bartolomeo, Sara and Angelini, Marco},
title = {{Weaving (Bio)Fabric: A Benchmark for Scalability Through Progressive Sampling Methods}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {23:1--23:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.23},
URN = {urn:nbn:de:0030-drops-279181},
doi = {10.4230/LIPIcs.GD.2026.23},
annote = {Keywords: Progressive visualization, Biofabric, Layout, Alternative Network Visualization}
}
Document
Using Reinforcement Learning to Optimize the Global and Local Crossing Number
Abstract
Graph drawing concerns the algorithmic visualization of graphs. A good drawing of a graph is easy to read and facilitates solving tasks on the graph. Several properties have been identified to occur in good drawings of graphs. Such properties include a low number of crossings, large angles between edges, short edges, and depicting symmetries. Many of these properties are explicitly measurable metrics. This lets us model a graph-drawing problem as a game where a single player iteratively moves vertices of a straight-line graph drawing to reduce edge crossings. We investigate whether reinforcement learning can discover effective strategies for playing this game. Our reinforcement-learning agent observes the local geometric and structural context of a vertex and selects a movement direction with the goal of reducing either the global or the local crossing number, that is, either the total number of crossings or the maximum number of crossings per edge. We compare the resulting strategies to existing methods and established crossing-minimization heuristics on standard benchmark graphs. While our approach does not out-compete state-of-the-art methods for minimizing the global crossing number, it is competitive and often superior for minimizing the local crossing number.
Cite as
Timo Brand, Henry Förster, Stephen Kobourov, Daniel Kohrt, Robin Schukrafft, Markus Wallinger, and Johannes Zink. Using Reinforcement Learning to Optimize the Global and Local Crossing Number. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brand_et_al:LIPIcs.GD.2026.24,
author = {Brand, Timo and F\"{o}rster, Henry and Kobourov, Stephen and Kohrt, Daniel and Schukrafft, Robin and Wallinger, Markus and Zink, Johannes},
title = {{Using Reinforcement Learning to Optimize the Global and Local Crossing Number}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {24:1--24:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.24},
URN = {urn:nbn:de:0030-drops-279193},
doi = {10.4230/LIPIcs.GD.2026.24},
annote = {Keywords: Reinforcement Learning, Crossing Minimization, Local Crossing Number}
}
Document
Does Graph Structure Affect Drawing Quality?
Abstract
Graph drawing aesthetic metrics (angular resolution, edge crossings, stress, and so on) quantify the visual quality of a graph drawing, and recent work has characterised their distributions across large drawing corpora. That analysis takes no account of the structure of the source graphs: the variation between graphs (driven by structural properties of the topology) is collapsed, together with the variation within a graph (driven by the choice of layout algorithm) into a single distribution per metric. This paper accounts for graph structure: for a fixed graph G with specified structural properties, what region of the eleven-dimensional metric-space is reachable, and how does that region depend on the structural properties?
We construct a stratified corpus of 89,493 graphs spanning generated families, established benchmarks, real-world networks, and curator-laid-out graphs from the literature, and lay each out with 18 algorithms drawn from the force-directed, layered, orthogonal, and structural families. Each graph’s reachable region is summarised by the minimum, maximum, mean, and range of 11 aesthetic metric values across its drawings. We then analyse drawing-level metric values using structural properties together with a layout-family encoding.
This analysis allows us to determine the extent to which a graph’s structural properties and the choice of layout family predict readability metrics, and to identify where graph-level metric ranges suggest structural limits on attainable layout quality. These results are important in setting expectations for graph drawing quality, for optimisation approaches, and for choosing graphs for which a specified layout quality is desirable.
Cite as
Gavin J. Mooney, Michael Wybrow, and Helen C. Purchase. Does Graph Structure Affect Drawing Quality?. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mooney_et_al:LIPIcs.GD.2026.25,
author = {Mooney, Gavin J. and Wybrow, Michael and Purchase, Helen C.},
title = {{Does Graph Structure Affect Drawing Quality?}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {25:1--25:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.25},
URN = {urn:nbn:de:0030-drops-279203},
doi = {10.4230/LIPIcs.GD.2026.25},
annote = {Keywords: Graph structure, Graph drawing, Metric landscape, Readability metrics}
}
Document
Opacity-Modulated Edge Drawing
Abstract
Edge crossings are a major source of visual clutter in node-link diagrams, causing visual interference through edge-segment overlaps. For graphs with fixed layouts, various approaches have been proposed to reduce crossings or their effects, including partial edge-drawing methods and techniques that modify edge shapes or routes. However, these approaches require changes to edge geometry or drawing extent. In this paper, we propose Opacity-Modulated Edge Drawing (OMED), a method to reduce interference while preserving edge geometry. OMED temporally modulates the opacity of each edge and controls the timing based on crossing relationships between edges, suppressing situations in which crossing edges simultaneously become highly visible. The method is formulated as a scheduling problem over crossing relationships between edges. To provide a unified evaluation framework for static and dynamic edge-drawing methods, we define three metrics: Degree of Visibility, Degree of Completeness, and Degree of Interference. Experimental evaluation based on these metrics showed that the max-fill implementation of OMED exhibited a different balance among visibility ratio, completeness ratio, and interference degree compared with representative existing edge-drawing methods. In particular, OMED (max-fill) maintained a visibility ratio comparable to that of the morphing edge drawing approach, while achieving a higher completeness ratio and a lower interference degree. These results demonstrate that different grouping strategies provide different trade-offs among visibility, completeness, and interference. This study therefore introduces a new approach based on temporal control of simultaneous visibility of crossing edges and expands the design space of edge-drawing methods.
Cite as
Kazuo Misue and Naoto Kikuta. Opacity-Modulated Edge Drawing. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{misue_et_al:LIPIcs.GD.2026.26,
author = {Misue, Kazuo and Kikuta, Naoto},
title = {{Opacity-Modulated Edge Drawing}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {26:1--26:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.26},
URN = {urn:nbn:de:0030-drops-279219},
doi = {10.4230/LIPIcs.GD.2026.26},
annote = {Keywords: node-link diagrams, edge drawing, visual clutter, opacity modulation, temporal visualization}
}
Document
Weighted Book Thickness
Abstract
We introduce and study the weighted book thickness of graphs. A k-page book embedding of a graph G = (V,E) is defined by a spanning cycle C for V (which does not need to be part of G) and a partition E = ⋃_{i=1}^k E_i such that E∩ C ⊆ E₁ and each graph G_i = (V,E_i∪ C), for 1 ≤ i ≤ k, is outerplane with outer cycle C. If e ∈ E_i, we say that e appears on Page i. The classical book thickness of a graph G is the minimum k such that there exists a k-page book embedding of G, that is, the minimum (over all book embeddings of G) achievable maximum page an edge appears on. In contrast, the weighted book thickness is the minimum achievable average page an edge appears on. The embeddings that realize weighted book thickness can differ from those that realize (classical) book thickness. We show that, although every planar graph on at most nine vertices admits a 2-page book embedding realizing its weighted book thickness, already for ten vertices, there is a planar graph for which every realization of its weighted book thickness needs more pages than its book thickness. We prove that there even exists a 2-tree whose weighted book thickness cannot be realized on two pages. On the positive side, we show that for every graph of pathwidth at most two, the weighted book thickness can always be realized by a 2-page book embedding and such an embedding can be found in linear time. Moreover, we prove that it is NP-complete to decide if the weighted book thickness is at most k, for some given integer k.
Cite as
Henry Förster, Michael Hoffmann, Stephen Kobourov, Maria Eleni Pavlidi, Alexandra Weinberger, and Johannes Zink. Weighted Book Thickness. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{forster_et_al:LIPIcs.GD.2026.27,
author = {F\"{o}rster, Henry and Hoffmann, Michael and Kobourov, Stephen and Pavlidi, Maria Eleni and Weinberger, Alexandra and Zink, Johannes},
title = {{Weighted Book Thickness}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {27:1--27:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.27},
URN = {urn:nbn:de:0030-drops-279228},
doi = {10.4230/LIPIcs.GD.2026.27},
annote = {Keywords: book thickness, linear layouts, 2-trees}
}
Document
Two Results on Outer-String Graphs
Abstract
An outer-string representation of a graph G is an intersection representation of G where vertices are represented by curves (strings) inside the unit disk, and each curve has exactly one endpoint on the boundary of the unit disk (the anchor of the curve). Additionally, if any two curves are allowed to cross at most once, we call this an outer-1-string representation of G. If we impose a cyclic ordering on the vertices of G and require the cyclic order of the anchors to respect this cyclic order, such a representation is called a constrained outer-string representation. In this paper, we present two results about graphs admitting outer-string representations.
Firstly, we show that for a bipartite graph G (and, more generally, for any {C₃,C₅}-free graph G) with a given cyclic order of vertices, we can decide in polynomial time whether G admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration, similar to that of Biedl et al. [GD 2024] for chordal graphs. Secondly, we answer an open question from the same authors and show that determining whether a given graph admits an outer-1-string representation is NP-hard. More generally, we show that it is NP-hard to determine if a given graph G admits an outer-k-string representation for any fixed k ≥ 1.
Cite as
Todor Antić, Vít Jelínek, Jan Kratochvíl, and Peter Stumpf. Two Results on Outer-String Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{antic_et_al:LIPIcs.GD.2026.28,
author = {Anti\'{c}, Todor and Jel{\'\i}nek, V{\'\i}t and Kratochv{\'\i}l, Jan and Stumpf, Peter},
title = {{Two Results on Outer-String Graphs}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.28},
URN = {urn:nbn:de:0030-drops-279232},
doi = {10.4230/LIPIcs.GD.2026.28},
annote = {Keywords: String graph, Outer-string graph, Constrained outer-string graph, Outer-1-string graph, Computational complexity, NP-hardness}
}
Document
Circle Graph 3-Coloring: Can Simplicity Beat SAT?
Abstract
We study the practical performance of the very recent quasi-polynomial algorithm for Circle Graph 3-Coloring (SOSA 2026 best paper), a problem of significant relevance to graph drawing due to its close connection to 3-page book embeddings. We implement and optimize this combinatorial algorithm and benchmark it against state-of-the-art SAT-based approaches on three sets of instances, the hardest of which required a new generation procedure to obtain non-trivial benchmarks. Across all three, our implementation outperforms the SAT-based approaches, with the largest gap on the most challenging instances.
Cite as
Robert Ganian, Philipp Mauve, and Vaishali Surianarayanan. Circle Graph 3-Coloring: Can Simplicity Beat SAT?. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 29:1-29:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ganian_et_al:LIPIcs.GD.2026.29,
author = {Ganian, Robert and Mauve, Philipp and Surianarayanan, Vaishali},
title = {{Circle Graph 3-Coloring: Can Simplicity Beat SAT?}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {29:1--29:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.29},
URN = {urn:nbn:de:0030-drops-279243},
doi = {10.4230/LIPIcs.GD.2026.29},
annote = {Keywords: Circle graphs, 3-coloring, book embeddings, simplicity in algorithms}
}
Document
Fast UMAP-Based Graph Drawing
Abstract
We present three fast UMAP-based graph drawing algorithms that reduce the time complexity of naive GUMAP from O(nm) to O(n² log n) and to O(n). Specifically, we reduce the time complexity for three components: (C0) O(nm)-time computation of all-pairs shortest paths, (C1) O(n² log n)-time computation of the kNN, and (C2) O(n)-time optimization of the cost function.
First, we present O(n² log n)-time SS-GUMAP, which utilizes spectral sparsification to compute a subgraph G' preserving structural properties of a graph G. Next, we present O(n)-time SL-GUMAP, which utilizes O(n)-time partial BFS to compute the shortest paths and kNN for C0 and C1, and sublinear-time O(n^0.9) edge sampling to reduce the cost optimization iterations for C2. Finally, we present a faster O(n)-time SSSL-GUMAP combining both approaches (i.e., running SL-GUMAP on the spectral sparsification G' of G) to reduce C0 and C1 to O(n) time and C2 to sublinear time, with faster runtime than SL-GUMAP in execution. Extensive experiments demonstrate that SS-GUMAP runs 28% faster than the baseline GUMAP, and SL-GUMAP and SSSL-GUMAP run 80% and 83% faster, respectively, while maintaining similar quality metrics and visual quality.
Cite as
Amyra Meidiana and Seok-Hee Hong. Fast UMAP-Based Graph Drawing. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 30:1-30:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{meidiana_et_al:LIPIcs.GD.2026.30,
author = {Meidiana, Amyra and Hong, Seok-Hee},
title = {{Fast UMAP-Based Graph Drawing}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {30:1--30:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.30},
URN = {urn:nbn:de:0030-drops-279257},
doi = {10.4230/LIPIcs.GD.2026.30},
annote = {Keywords: Graph Drawing, UMAP, Spectral Sparsification}
}
Document
1-Planar Drawings of Products of Cycles
Abstract
We show that the Cartesian product of two cycles C_m × C_n, i.e. the m × n toroidal grid, admits exactly three distinct simple 1-planar drawings when 5 ≤ m ≤ n. These drawings have (m-2)n, m(n-2), and mn - 4 crossings, respectively. As a corollary, we obtain that the 1-planar crossing number of C_m × C_n with 3 ≤ m ≤ n is (m-2)n. This confirms, in the 1-planar setting, a conjecture of Harary, Kainen, and Schwenk. The drawings have the property that the n fundamental cycles C_m and the m fundamental cycles C_n of C_m× C_n form two classes of pairwise disjoint cycles. In fact, establishing this property is a crucial step in our proof.
Cite as
Stefan Felsner, Kolja Knauer, Yasumi Kogo, and Bobby Miraftab. 1-Planar Drawings of Products of Cycles. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{felsner_et_al:LIPIcs.GD.2026.31,
author = {Felsner, Stefan and Knauer, Kolja and Kogo, Yasumi and Miraftab, Bobby},
title = {{1-Planar Drawings of Products of Cycles}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.31},
URN = {urn:nbn:de:0030-drops-279267},
doi = {10.4230/LIPIcs.GD.2026.31},
annote = {Keywords: 1-planar, drawings, cycles, product, torus}
}
Document
On the 2-Bend Slope Number of 1-Planar Graphs
Abstract
While drawing planar graphs with few slopes and few bends is a well-studied problem, corresponding extensions to beyond-planar graphs still remain mostly unexplored. Motivated by this observation, in this work, we provide bounds on the slope number of biconnected 1-planar graphs when two bends are allowed along each edge. Our contribution is an incremental drawing algorithm that produces 2-bend 1-planar drawings of biconnected 1-plane graphs with maximum degree Δ using any prescribed set of Δ pairwise distinct slopes.
Cite as
Michael A. Bekos, Eleni Katsanou, Philipp Kindermann, Aikaterini Maria Ntasiou, Maria Eleni Pavlidi, and Soeren Terziadis. On the 2-Bend Slope Number of 1-Planar Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bekos_et_al:LIPIcs.GD.2026.32,
author = {Bekos, Michael A. and Katsanou, Eleni and Kindermann, Philipp and Ntasiou, Aikaterini Maria and Pavlidi, Maria Eleni and Terziadis, Soeren},
title = {{On the 2-Bend Slope Number of 1-Planar Graphs}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {32:1--32:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.32},
URN = {urn:nbn:de:0030-drops-279279},
doi = {10.4230/LIPIcs.GD.2026.32},
annote = {Keywords: k-bend drawings, 1-planar graphs, beyond planarity, slope number}
}
Document
A First View on the Density of 5-Planar Graphs
Abstract
A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to k-planar graphs, which can be drawn in the plane with at most k > 0 crossings per edge. One of the basic graph properties that have been explored for those graph classes is the maximum edge density, i.e., the maximum number of edges a k-planar graph on n vertices may have. While there are numerous results for the classes of 1- and 2-planar graphs, there are few results for increasing k = 3 or 4 due to the complex graph structures. We make a first step towards even larger k > 4 exploring the class of 5-planar graphs. While our main tool is still a discharging technique, a better understanding of the structure of the denser parts leads to corresponding density bounds in a much simpler way.
We first apply a simplified version of our technique to outer 5-planar graphs and surprisingly observe that the structure of maximally dense (general) 5-planar graphs differs from the known uniform structure of maximally dense k-planar graphs for smaller k ∈ {1,2,3,4}. As the central result of this paper, we then show that graphs that admit a simple 5-planar drawing have at most 7(n-2) edges, drastically improving the previous best bound of ≈8.3n. This even implies a small improvement of the leading constant in the Crossing Lemma cr(G) ≥ c m³/n² from c = 1/(27.48) to c = 1/(27.3). To demonstrate the potential of our new technique, we also apply it to 4-planar and 6-planar graphs.
Cite as
Aaron Büngener, Jakob Franz, Michael Kaufmann, and Maximilian Pfister. A First View on the Density of 5-Planar Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bungener_et_al:LIPIcs.GD.2026.33,
author = {B\"{u}ngener, Aaron and Franz, Jakob and Kaufmann, Michael and Pfister, Maximilian},
title = {{A First View on the Density of 5-Planar Graphs}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {33:1--33:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.33},
URN = {urn:nbn:de:0030-drops-279286},
doi = {10.4230/LIPIcs.GD.2026.33},
annote = {Keywords: k-planar graphs, 5-planarity, edge density, Crossing Lemma, discharging}
}
Document
On t-Colorable k-Plane Drawings
Abstract
In this work, we introduce t-colorable k-plane drawings, that is, drawings of graphs with a t-edge-coloring where every edge is crossed by at most k edges of each color. We give tight upper bounds on the edge- and crossing density for small values of t and k and show that the recognition of such drawings is NP-complete if t ≥ 2 and k ≥ 1.
Cite as
Miriam Goetze, Michael Kaufmann, and Soeren Terziadis. On t-Colorable k-Plane Drawings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{goetze_et_al:LIPIcs.GD.2026.34,
author = {Goetze, Miriam and Kaufmann, Michael and Terziadis, Soeren},
title = {{On t-Colorable k-Plane Drawings}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {34:1--34:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.34},
URN = {urn:nbn:de:0030-drops-279298},
doi = {10.4230/LIPIcs.GD.2026.34},
annote = {Keywords: beyond planar graphs, k-planarity, edge density, crossing density, recognition problem, NP-completeness}
}
Document
Point Set Embeddability with List Constraints
Abstract
Deciding whether a given graph admits a planar straight-line drawing where each vertex is placed on some point from a given finite point set is known as Point Set Embeddability and is a classical problem in graph drawing. In this paper, we study the more general embeddability question where the placement of each vertex v is restricted to a list L(v) of admissible points.
We first study the case where the given point set is in convex position. We show that this case is NP-hard even if the given graph is a matching and bi-labeled, i.e., each vertex has at most 2 admissible points. On the positive side, we present two efficient algorithms for the case where the given graph G is connected (and not necessarily bi-labeled): if G is equipped with a combinatorial embedding that needs to be respected, we can solve the problem in polynomial time; otherwise we can solve it in FPT-time with regard to the maximum vertex degree. In particular, this answers an open question by Frati, Glisse, Lenhart, Liotta, Mchedlidze, and Nishat [GD'13].
We then turn our attention to the more general case where the given point set is not necessarily in convex position. Here, we show NP-hardness for bi-labeled paths; notably these graphs have a unique combinatorial embedding and maximum degree two, thereby creating a stark contrast to our algorithms for the convex case. We also present an FPT-algorithm with respect to the vertex cover number for the special case of bi-labeled graphs. We complement this latter result by establishing paraNP-hardness in the tri-labeled setting for vertex cover number 2 and polynomial-time solvability for vertex cover number 1 and arbitrary L.
Finally, we study optimization and extension variants, where we want to maximize the number of edges or extend a partial drawing, respectively. For the former, we show APX-hardness and for the latter, we provide a parameterized complexity dichotomy under natural extension parameters.
Cite as
Thomas Depian, Joseph Dorfer, Boris Klemz, Matthias Pfretzschner, and Lena Schlipf. Point Set Embeddability with List Constraints. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{depian_et_al:LIPIcs.GD.2026.35,
author = {Depian, Thomas and Dorfer, Joseph and Klemz, Boris and Pfretzschner, Matthias and Schlipf, Lena},
title = {{Point Set Embeddability with List Constraints}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {35:1--35:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.35},
URN = {urn:nbn:de:0030-drops-279308},
doi = {10.4230/LIPIcs.GD.2026.35},
annote = {Keywords: Point set embedding, List constraints, Parameterized algorithms and complexity, NP-hardness, Dynamic programming, Graph drawing, Computational geometry}
}
Document
Geometric (1+ε)-Spanners with Few Crossings
Abstract
For n points in the plane and an ε > 0, we construct a (1+ε)-spanner with O(n/ε) edges in which every edge has Õ(1/ε³) crossings, hence the total number of crossings is Õ(n/ε⁴), furthermore the ratio between the lengths of any two crossing edges is O(1/ε²). Our spanner construction substantially improves on the previous upper bound for the number of crossings in a (1+ε)-spanner, and it is the first spanner construction that ensures O(1) crossings per edge for any constant ε > 0. In contrast, we construct: n points in the plane for which every (1+ε)-spanner has Ω(n/ε³) crossings, n points for which every (1+ε)-spanner has an edge with Ω(1/ε^{5/2}) crossings, and 4 points for which every (1+ε)-spanner contains two crossing edges where one is Ω(1/ε) times longer than the other.
Cite as
Kelvin Luu and Csaba D. Tóth. Geometric (1+ε)-Spanners with Few Crossings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{luu_et_al:LIPIcs.GD.2026.36,
author = {Luu, Kelvin and T\'{o}th, Csaba D.},
title = {{Geometric (1+\epsilon)-Spanners with Few Crossings}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {36:1--36:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.36},
URN = {urn:nbn:de:0030-drops-279312},
doi = {10.4230/LIPIcs.GD.2026.36},
annote = {Keywords: crossings, spanners, empty region property}
}
Document
Structural Properties of Shortest Flip Sequences Between Plane Spanning Trees
Abstract
We study the problem of reconfiguring plane spanning trees on point sets in the plane in convex position. A flip is a reconfiguration step in which one edge of a tree is replaced with another, yielding again a plane spanning tree. For a given point set P, the flip graph has a vertex for each plane spanning tree on P, and an edge between any two trees that differ by a single flip. The flip distance between two trees is the length of a shortest path between them in the flip graph, that is, the minimum number of flips needed to transform one tree into the other.
We study structural properties of shortest flip sequences. Over the years, several conjectures have been proposed regarding these sequences. The folklore happy edge conjecture suggests that any edge shared by both the initial and target tree is never flipped in a shortest flip sequence. Moreover, the parking conjecture claims that in shortest flip sequences, every edge is either flipped to a target edge or parked on the convex hull. Notably, the happy edge conjecture is implied by the parking conjecture. Furthermore, it has been conjectured that an edge is never reparked, that is, any edge resulting from a flip either is a target edge or is subsequently flipped to a target edge. Essentially, all recent flip algorithms rely on these three conjectures and the properties they imply.
In this paper, we disprove both the parking and the reparking conjecture. In particular, we show that a flip sequence obeying the parking property sometimes needs linearly more flips than a shortest flip sequence, and exhibit examples where any shortest flip sequence flips some edge multiple times. In contrast, we identify several conditions under which the parking and reparking properties hold; these imply that the reparking property holds for compatible flips, and may be exploited by algorithms to obtain short flip sequences, especially for the compatible case.
Cite as
Oswin Aichholzer, Joseph Dorfer, Peter Kramer, Christian Rieck, and Birgit Vogtenhuber. Structural Properties of Shortest Flip Sequences Between Plane Spanning Trees. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aichholzer_et_al:LIPIcs.GD.2026.37,
author = {Aichholzer, Oswin and Dorfer, Joseph and Kramer, Peter and Rieck, Christian and Vogtenhuber, Birgit},
title = {{Structural Properties of Shortest Flip Sequences Between Plane Spanning Trees}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {37:1--37:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.37},
URN = {urn:nbn:de:0030-drops-279323},
doi = {10.4230/LIPIcs.GD.2026.37},
annote = {Keywords: Spanning trees, convex position, non-crossing, flip graph, happy edges, parking conjecture}
}
Document
Graph Drawing Contest Report
Graph Drawing Contest Report (Graph Drawing Contest Report)
Abstract
This report describes the challenges and winners of the 2026 Graph Drawing Contest, held in conjunction with the 34th International Symposium on Graph Drawing and Network Visualization (GD'26) at Brock University in St. Catharines, Ontario, Canada. The mission of the Graph Drawing Contest is to monitor and challenge the current state of the art in graph-drawing technology. This year’s edition featured two categories: a creative topic in which participants visualized voting patterns across the years of the Eurovision Song Contest, and a live challenge held at the conference, where participants had to produce straight-line drawings of graphs on a grid that were k-plane for as low a value of k as possible. The creative topic focused in particular on revealing patterns in how countries vote for one another over time, using a dynamic and multivariate dataset containing both jury and public votes.
Cite as
Sara Di Bartolomeo, Henry Förster, Soeren Terziadis, and Jules Wulms. Graph Drawing Contest Report (Graph Drawing Contest Report). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 38:1-38:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dibartolomeo_et_al:LIPIcs.GD.2026.38,
author = {Di Bartolomeo, Sara and F\"{o}rster, Henry and Terziadis, Soeren and Wulms, Jules},
title = {{Graph Drawing Contest Report}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {38:1--38:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.38},
URN = {urn:nbn:de:0030-drops-279334},
doi = {10.4230/LIPIcs.GD.2026.38},
annote = {Keywords: Graph Drawing, Information Visualization, Graph Drawing Contest}
}
Document
Graph Drawing Contest Abstract
Eurovision Song Contest Voting Communities (Graph Drawing Contest Abstract)
Abstract
We present our approach to tackle and win the Creative Topic of the 2026 Graph Drawing Contest, which focused on voting patterns in the Eurovision Song Contest. We describe our driving question, the data preprocessing and filtering, and the design of the poster, including the selected visualizations.
Cite as
Charan Kumar Kharade Somoji Rao, Melanie Baur, and Michael Krone. Eurovision Song Contest Voting Communities (Graph Drawing Contest Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 39:1-39:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{somojirao_et_al:LIPIcs.GD.2026.39,
author = {Somoji Rao, Charan Kumar Kharade and Baur, Melanie and Krone, Michael},
title = {{Eurovision Song Contest Voting Communities}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {39:1--39:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.39},
URN = {urn:nbn:de:0030-drops-279342},
doi = {10.4230/LIPIcs.GD.2026.39},
annote = {Keywords: Graph Drawing Contest 2026 Creative Topic: Eurovision Song Contest}
}
Document
Graph Drawing Contest Abstract
Optimizing k-Planarity (Graph Drawing Contest Abstract)
Abstract
An embedding of a graph is k-planar if every edge has at most k crossings. The 2026 Graph Drawing Challenge gave a one-hour time window to create minimal k-planar straight-line embeddings for 15 graphs.
I present the winning approach. Initial layouts come from several construction heuristics and are then optimized by a tuned local search or by simulated annealing. Both heuristics move one node at a time, so the rate at which candidate moves can be evaluated is crucial for performance. A uniform lookup grid restricts the computation of each node movement to the edges it can affect, and the evaluation aborts as soon as the move is known to fail. Together these two optimizations reach hundreds of thousands of proposals per second on all competition instances. That was enough to run every instance more than 20 times within the time limit, which let me take or tie the best score on all 15 graphs.
Cite as
Christoph Weber. Optimizing k-Planarity (Graph Drawing Contest Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 40:1-40:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{weber:LIPIcs.GD.2026.40,
author = {Weber, Christoph},
title = {{Optimizing k-Planarity}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {40:1--40:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.40},
URN = {urn:nbn:de:0030-drops-279354},
doi = {10.4230/LIPIcs.GD.2026.40},
annote = {Keywords: k-planarity, crossing minimization, local search, simulated annealing}
}
Document
Poster Abstract
Size Should Not Matter: Evaluating Network Visualizations with Stress (Poster Abstract)
Abstract
The normalized stress metric is widely used to assess graph drawing quality, measuring how closely distances between vertices in a layout match their graph-theoretic distances. This metric is a standard for both evaluation and optimization in many popular graph layout algorithms. However, normalized stress is highly sensitive to the scale (size) of the drawing, leading to potentially misleading comparisons between layouts produced by different algorithms. Uniformly scaling a layout can significantly alter stress values without changing the underlying structure, even to the extent that a clearly superior layout can appear to have a higher stress score than a random layout. Although this issue is recognized within the network visualization community, it is rarely addressed with sufficient detail in publications, resulting in critical calculation errors in recent studies. In this paper, we systematically examine various stress metrics used in the literature and demonstrate that commonly used metrics are affected by layout scale, undermining their reliability for comparison. We identify a scale-invariant alternative and propose scale-normalized stress for fair stress-based evaluation.
Cite as
Reyan Ahmed, Cesim Erten, Stephen Kobourov, Jonah Lotz, Jacob Miller, and Hamlet Taraz. Size Should Not Matter: Evaluating Network Visualizations with Stress (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 41:1-41:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ahmed_et_al:LIPIcs.GD.2026.41,
author = {Ahmed, Reyan and Erten, Cesim and Kobourov, Stephen and Lotz, Jonah and Miller, Jacob and Taraz, Hamlet},
title = {{Size Should Not Matter: Evaluating Network Visualizations with Stress}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {41:1--41:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.41},
URN = {urn:nbn:de:0030-drops-279363},
doi = {10.4230/LIPIcs.GD.2026.41},
annote = {Keywords: Normalized stress, layout evaluation, scale-invariant metrics}
}
Document
Poster Abstract
Hamiltonian Properties of Cycle-Trees (Poster Abstract)
Abstract
We investigate the Hamiltonian properties of cycle-trees. Using the SPQ-decomposition, we show that cycle-trees are sub-Hamiltonian and that 3-connected cycle-trees are Hamiltonian.
Cite as
Giordano Andreola, Giordano Da Lozzo, and Marco D'Elia. Hamiltonian Properties of Cycle-Trees (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 42:1-42:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{andreola_et_al:LIPIcs.GD.2026.42,
author = {Andreola, Giordano and Da Lozzo, Giordano and D'Elia, Marco},
title = {{Hamiltonian Properties of Cycle-Trees}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {42:1--42:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.42},
URN = {urn:nbn:de:0030-drops-279375},
doi = {10.4230/LIPIcs.GD.2026.42},
annote = {Keywords: Cycle-trees, SPQ-trees, Halin graphs, Hamiltonicity}
}
Document
Poster Abstract
On the Recognition of Outerplanar Graphs with Queue Number 1 (Poster Abstract)
Abstract
A linear layout of a graph is defined as a total order of the vertices and a partition of the edges to pages. In a stack (queue) layout, no two edges on the same page may cross (nest). The stack (queue) number of a graph is the minimum number of pages required in a stack (queue) layout. Our focus is on characterizing and recognizing graphs that have both stack number 1 and queue number 1.
Cite as
Michael A. Bekos, Thomas Depian, Stefan Felsner, Michael Kaufmann, Philipp Kindermann, Fabrizio Montecchiani, Maria Eleni Pavlidi, Alexandra Weinberger, Alexander Wolff, and Johannes Zink. On the Recognition of Outerplanar Graphs with Queue Number 1 (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 43:1-43:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bekos_et_al:LIPIcs.GD.2026.43,
author = {Bekos, Michael A. and Depian, Thomas and Felsner, Stefan and Kaufmann, Michael and Kindermann, Philipp and Montecchiani, Fabrizio and Pavlidi, Maria Eleni and Weinberger, Alexandra and Wolff, Alexander and Zink, Johannes},
title = {{On the Recognition of Outerplanar Graphs with Queue Number 1}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {43:1--43:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.43},
URN = {urn:nbn:de:0030-drops-279381},
doi = {10.4230/LIPIcs.GD.2026.43},
annote = {Keywords: stack layouts, queue layouts, outerplanar, outerpaths}
}
Document
Poster Abstract
Toroidal Graph Drawing by Multidimensional Scaling (Poster Abstract)
Abstract
We present a framework for multidimensional scaling of graph data to the flat torus. Compared to prior approaches, we contribute a closed-form geodesic distance computation, and learnable parameters for the torus size and aspect ratio, with an extension to rhombic tori. On synthetic benchmarks, our layouts achieve lower normalized stress than Euclidean baselines, and on real toroidal data our method recovers the underlying structure.
Cite as
Timo Brand, Sebastian Damrich, Stephen Kobourov, and Jacob Miller. Toroidal Graph Drawing by Multidimensional Scaling (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 44:1-44:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brand_et_al:LIPIcs.GD.2026.44,
author = {Brand, Timo and Damrich, Sebastian and Kobourov, Stephen and Miller, Jacob},
title = {{Toroidal Graph Drawing by Multidimensional Scaling}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {44:1--44:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.44},
URN = {urn:nbn:de:0030-drops-279393},
doi = {10.4230/LIPIcs.GD.2026.44},
annote = {Keywords: Graph layout, non-Euclidean geometry, multidimensional scaling}
}
Document
Poster Abstract
Newey: An Algorithmically-Assisted Editor for High-Quality Fully Labelled Transit Map Layout (Poster Abstract)
Abstract
Transit maps visualize geospatial graphs such as metro-, tram- and bus-networks, and are canonically schematized using octolinear layouts. Recently, Van Dijk and Terziadis [van Dijk and Terziadis, 2026] released a graphical user interface for schematic transit map design that provides the user with a compelling set of algorithmically-assisted interactions based on "port assignments" - sidestepping the issue of computational complexity and including a human in the loop for aesthetic judgement. As is often the case for automated transit map design systems, their editor focuses solely on the graph layout problem without considering the need to actually label the stations. In this poster we demonstrate a significantly more capable editor that increases usability and practical relevance in two ways. First, we demonstrate that labelling can be seamlessly integrated in the port assignment framework, with efficient and meaningful algorithmic assistance. Secondly, we introduce various novel ways of interacting with the port assignment, providing several intuitive user interactions within the framework (for example: dragging nodes, and scaling and pivoting selections).
Cite as
Morris Boers, Thomas C. van Dijk, and Soeren Terziadis. Newey: An Algorithmically-Assisted Editor for High-Quality Fully Labelled Transit Map Layout (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 45:1-45:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{boers_et_al:LIPIcs.GD.2026.45,
author = {Boers, Morris and van Dijk, Thomas C. and Terziadis, Soeren},
title = {{Newey: An Algorithmically-Assisted Editor for High-Quality Fully Labelled Transit Map Layout}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {45:1--45:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.45},
URN = {urn:nbn:de:0030-drops-279403},
doi = {10.4230/LIPIcs.GD.2026.45},
annote = {Keywords: algorithms, schematic map design, transit maps, labeling, human in the loop}
}
Document
Poster Abstract
Circular Outside-Obstacle Representations of 2-Trees (Poster Abstract)
Abstract
An outside-obstacle representation (OOR) of a graph G places its vertices in the plane together with a single obstacle in the outer face such that two vertices are adjacent if and only if the segment between them avoids the obstacle. An OOR is reducible if every edge is incident to the outer face, which implies that every subgraph also admits an OOR. Resolving a conjecture of Firman et al. [GD 2022], we show that every outerplanar graph admits a reducible circular OOR, that is, one with all vertices co-circular. In contrast, we present a family of 2-trees that admits no such representation.
Cite as
Stefan Felsner, Carolina Haase, Philipp Kindermann, and William J. Lenhart. Circular Outside-Obstacle Representations of 2-Trees (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 46:1-46:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{felsner_et_al:LIPIcs.GD.2026.46,
author = {Felsner, Stefan and Haase, Carolina and Kindermann, Philipp and Lenhart, William J.},
title = {{Circular Outside-Obstacle Representations of 2-Trees}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {46:1--46:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.46},
URN = {urn:nbn:de:0030-drops-279414},
doi = {10.4230/LIPIcs.GD.2026.46},
annote = {Keywords: Obstacle Representations, Outerplanar Graphs, 2-Trees}
}
Document
Poster Abstract
Chunky Chains: Graph Drawing on Narrow Screens (Poster Abstract)
Abstract
We introduce Chunky Chains, a graph drawing style with fixed width. A Chunky Chain consists of a chain of chord diagrams, called buckets, and is bounded by a given capacity c that limits the number of vertices in each bucket. Edges whose endpoints lie in the same or in adjacent buckets are short. Edges that are "skipping" at least one bucket are long, and we draw them only partially. The goal is to minimize the number of long edges.
We study the complexity of this problem, give exact and parameterized algorithms, and describe (and implement) a practical drawing pipeline.
Cite as
Tim Hegemann, Dominik Jilg, Marie Diana Sieper, and Samuel Wolf. Chunky Chains: Graph Drawing on Narrow Screens (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 47:1-47:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hegemann_et_al:LIPIcs.GD.2026.47,
author = {Hegemann, Tim and Jilg, Dominik and Sieper, Marie Diana and Wolf, Samuel},
title = {{Chunky Chains: Graph Drawing on Narrow Screens}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {47:1--47:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.47},
URN = {urn:nbn:de:0030-drops-279428},
doi = {10.4230/LIPIcs.GD.2026.47},
annote = {Keywords: Hybrid visualization, chord-link, chord diagram, chunky chains, bucketwidth, bandwidth, bucket integrity, experimental analysis}
}
Document
Poster Abstract
Independence of the Lower Bound for Geometric Triangulation Drawings on the Number of Chains (Poster Abstract)
Abstract
In 2026, Cruces, Huemer, and Lara established that a specific combinatorial triangulation can be drawn in at least Ω(3^{n/4}) ways on the double chain as geometric triangulation. We generalize their construction to a d-tuple chain configuration. By introducing the switch subgraph abstraction to count recursive drawing, we obtain the same asymptotic lower bounds Ω(3^{n/4}) ≈ Ω(1.316ⁿ) even when we increase the number of chains.
Cite as
Vojtěch Vincíbr, Anežka Kasalová, Zuzana Reiffersová, and Petr Filip. Independence of the Lower Bound for Geometric Triangulation Drawings on the Number of Chains (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 48:1-48:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vincibr_et_al:LIPIcs.GD.2026.48,
author = {Vinc{\'\i}br, Vojt\v{e}ch and Kasalov\'{a}, Ane\v{z}ka and Reiffersov\'{a}, Zuzana and Filip, Petr},
title = {{Independence of the Lower Bound for Geometric Triangulation Drawings on the Number of Chains}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {48:1--48:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.48},
URN = {urn:nbn:de:0030-drops-279435},
doi = {10.4230/LIPIcs.GD.2026.48},
annote = {Keywords: combinatorial triangulation, geometric triangulation, planar graph, d-tuple chain, switch subgraph}
}
Document
Poster Abstract
RACTrix: Interactive Visual Summaries for Exploring Signed Networks (Poster Abstract)
Abstract
Force-directed network layouts naturally reveal densely connected subgraphs or communities as distinct spatial clusters. However, since node positions are used to encode the graph structure, other node and edge attributes must be represented through secondary visual encodings such as color, shape, or edge thickness. This makes it difficult to accurately interpret quantitative information such as node rankings, community sizes, and edge properties without additional interactions, such as hovering, clicking, or issuing queries. The challenges become even more severe for signed graphs, where edges can be weighted and carry either positive or negative labels. Here we present RACTrix, a compact, intuitive, and interactive visual summary to aid in the exploration of signed networks. RACTrix combines the aesthetics of right-angle crossing (RAC) drawings with a matrix-like representation and links them with a force-directed layout. RACTrix provides positional encoding for node/cluster rankings, edge rankings around a node/cluster, edge signs, edge directions, and cluster sizes, while it maintains a matrix-based representation of pairwise relationships among nodes/clusters. We coordinate RACTrix and force-directed layout such that selecting an element in one view highlights the corresponding elements in the other. This allows for exploring quantitative network features alongside the network structure which is crucial for validating visual patterns in the network layout.
Cite as
Dante Tostado Cortes and Debajyoti Mondal. RACTrix: Interactive Visual Summaries for Exploring Signed Networks (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 49:1-49:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tostadocortes_et_al:LIPIcs.GD.2026.49,
author = {Tostado Cortes, Dante and Mondal, Debajyoti},
title = {{RACTrix: Interactive Visual Summaries for Exploring Signed Networks}},
booktitle = {34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
pages = {49:1--49:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-452-9},
ISSN = {1868-8969},
year = {2026},
volume = {396},
editor = {L\"{o}ffler, Maarten and Miksch, Silvia},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.49},
URN = {urn:nbn:de:0030-drops-279441},
doi = {10.4230/LIPIcs.GD.2026.49},
annote = {Keywords: Network visualization, Interactive summary, RAC drawing, Signed graph}
}