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String graph with Cop Number 4 32

String Graph with Cop Number 4

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Durocher, Stephane Kryven, Myroslav Löffler, Maarten Department of Computer Science University of Manitoba Winnipeg Canada Department of Information and Computing Sciences Utrecht University Netherlands

ISBN: (纸本)9783959773430

Cops and Robbers is a well-studied pursuit-evasion game in which a set of cops seeks to catch a robber in a graph G, where cops and the robber move along edges of G. The cop number of G is the minimum number of cops that is sufficient to catch the robber. The game of Cops and Robbers has been well-studied on beyond-planar graphs (that is, graphs that can be drawn with only few crossings) [1, 4] as well as intersection graphs (that is, graphs where the vertices represent geometric objects, and an edge exists between two vertices if the corresponding objects intersect). We consider a well-known subclass of intersection graphs called string graphs where the objects are curves. So far no string graph with cop number larger than three was known. We construct the first string graph with cop number four, which improves the previous bound and answers an open question by Gavenčiak et al. [5]. © Stephane Durocher, Myroslav Kryven, and Maarten Löffler.

关键词： graph embeddings

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The Perception of Stress in graph drawings 32

The Perception of Stress in Graph Drawings

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Mooney, Gavin J. Purchase, Helen C. Wybrow, Michael Kobourov, Stephen G. Miller, Jacob Monash University Melbourne Australia Technical University of Munich Campus Heilbronn Germany

ISBN: (纸本)9783959773430

Most of the common graph layout principles (a.k.a. "aesthetics") on which many graph drawing algorithms are based are easy to define and to perceive. For example, the number of pairs of edges that cross each other, how symmetric a drawing looks, the aspect ratio of the bounding box, or the angular resolution at the nodes. The extent to which a graph drawing conforms to these principles can be determined by looking at how it is drawn - that is, by looking at the marks on the page - without consideration for the underlying structure of the graph. A key layout principle is that of optimising "stress", the basis for many algorithms such as the popular Kamada & Kawai algorithm and several force-directed algorithms. The stress of a graph drawing is, loosely speaking, the extent to which the geometric distance between each pair of nodes is proportional to the shortest path between them - over the whole graph drawing. The definition of stress therefore relies on the underlying structure of the graph (the "paths") in a way that other layout principles do not, making stress difficult to describe to novices unfamiliar with graph drawing principles, and, we believe, difficult to perceive. We conducted an experiment to see whether people (novices as well as experts) can see stress in graph drawings, and found that it is possible to train novices to "see"stress - even if their perception strategies are not based on the definitional concepts. © Gavin J. Mooney, Helen C. Purchase, Michael Wybrow, Stephen G. Kobourov, and Jacob Miller.

关键词： graph algorithms

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Monotone Arc Diagrams with Few Biarcs 32

Monotone Arc Diagrams with Few Biarcs

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Chaplick, Steven Förster, Henry Hoffmann, Michael Kaufmann, Michael Maastricht University Netherlands Universität Tübingen Germany Department of Computer Science ETH Zürich Switzerland

ISBN: (纸本)9783959773430

We show that every planar graph has a monotone topological 2-page book embedding where at most (4n - 10)/5 (of potentially 3n - 6) edges cross the spine, and every edge crosses the spine at most once;such an edge is called a biarc. We can also guarantee that all edges that cross the spine cross it in the same direction (e.g., from bottom to top). For planar 3-trees we can further improve the bound to (3n - 9)/4, and for so-called Kleetopes we obtain a bound of at most (n - 8)/3 edges that cross the spine. The bound for Kleetopes is tight, even if the drawing is not required to be monotone. A Kleetope is a plane triangulation that is derived from another plane triangulation T by inserting a new vertex vf into each face f of T and then connecting vf to the three vertices of f. © Steven Chaplick, Henry Förster, Michael Hoffmann, and Michael Kaufmann.

关键词： graph embeddings

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Note on Min-k-Planar drawings of graphs 32

Note on Min-k-Planar Drawings of Graphs

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Hliněný, Petr Ködmön, Lili Masaryk University Brno Czech Republic Eötvös Loránd University Budapest Hungary

ISBN: (纸本)9783959773430

The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., gd 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple. In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k ≤ 3 known (but perhaps not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k ≥ 4, the difference between general and simple drawings in the latter concept is more striking. We prove that there exist graphs with a min-2-planar drawing, or with a min-3-planar drawing avoiding crossings of adjacent edges, which have no simple min-k-planar drawings for arbitrarily large fixed k. © Petr Hliněný and Lili Ködmön.

关键词： graph theory

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Quantum Algorithms for One-Sided Crossing Minimization 32

Quantum Algorithms for One-Sided Crossing Minimization

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Caroppo, Susanna Da Lozzo, Giordano Di Battista, Giuseppe Roma Tre University Rome Italy

ISBN: (纸本)9783959773430

We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. We show that OSCM can be viewed as a set problem amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. SODA 2019] to devise a QRAM-based algorithm that solves OSCM in O∗(1.728n) time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in O∗(2n) time and polynomial space. © Susanna Caroppo, Giordano Da Lozzo, and Giuseppe Di Battista.

关键词： exact exponential algorithms One-sided crossing minimization quantum divide and conquer quantum dynamic programming quantum graph drawing

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1-Planar Unit Distance graphs 32

1-Planar Unit Distance Graphs

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Gehér, Panna Tóth, Géza Eötvös Loránd University Budapest Hungary Alfréd Rényi Institute of Mathematics Budapest Hungary

ISBN: (纸本)9783959773430

A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on n vertices can have at most [3n - √12n - 3] edges. Recently his conjecture was settled by Lavollée and Swanepoel. In this paper we consider 1-planar unit distance graphs. We say that a graph is a 1-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on n vertices can have at most 3n - 4 √n/10 edges. © Panna Gehér and Géza Tóth.

关键词： graph theory

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graphTrials: Visual Proofs of graph Properties 32

GraphTrials: Visual Proofs of Graph Properties

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Förster, Henry Klesen, Felix Dwyer, Tim Eades, Peter Hong, Seok-Hee Kobourov, Stephen G. Liotta, Giuseppe Misue, Kazuo Montecchiani, Fabrizio Pastukhov, Alexander Schreiber, Falk Department of Computer Science University of Tübingen Germany Institute of Computer Science University of Würzburg Germany Faculty of Information Technology Monash University Australia School of Computer Science The University of Sydney Australia Department of Computer Science University of Arizona TucsonAZ United States Department of Engineering University of Perugia Italy Department of Computer Science University of Tsukuba Japan Department of Psychology University of Bamberg Germany Department of Computer Science University of Konstanz Germany

ISBN: (纸本)9783959773430

graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a greater role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e. g. via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Finally, we provide examples of visual certificates for problems in different visual proof complexity classes. © Henry Förster, Felix Klesen, Tim Dwyer, Peter Eades, Seok-Hee Hong, Stephen G. Kobourov, Giuseppe Liotta, Kazuo Misue, Fabrizio Montecchiani, Alexander Pastukhov, and Falk Schreiber.

关键词： Data visualization

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On k-Plane Insertion into Plane drawings 32

On k-Plane Insertion into Plane Drawings

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Katheder, Julia Kindermann, Philipp Klute, Fabian Parada, Irene Rutter, Ignaz Universität Tübingen Germany Universität Trier Germany Universitat Politècnica de Catalunya Barcelona Spain Department of Mathematics Universitat Politècnica de Catalunya Barcelona Spain Universität Passau Germany

ISBN: (纸本)9783959773430

We introduce the k-Plane Insertion into Plane drawing (k-PIP) problem: given a plane drawing of a planar graph G and a set F of edges, insert the edges in F into the drawing such that the resulting drawing is k-plane. In this paper, we show that the problem is NP-complete for every k ≥ 1, even when G is biconnected and the set F of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that k = 1 and G is a triangulation. © Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, and Ignaz Rutter.

关键词： drawing (graphics)

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Bundling-Aware graph drawing 32

Bundling-Aware Graph Drawing

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Archambault, Daniel Liotta, Giuseppe Nöllenburg, Martin Piselli, Tommaso Tappini, Alessandra Wallinger, Markus Newcastle University United Kingdom University of Perugia Italy TU Wien Austria TU Munich Germany

ISBN: (纸本)9783959773430

Edge bundling algorithms significantly improve the visualization of dense graphs by reducing the clutter of many edges visible on screen by bundling them together. As such, bundling is often viewed as a post-processing step applied to a drawing, and the vast majority of edge bundling algorithms consider a graph and its drawing as input. Another way of thinking about edge bundling is to simultaneously optimize both the drawing and the bundling. In this paper, we investigate methods to simultaneously optimize a graph drawing and its bundling. We describe an algorithmic framework which consists of three main steps, namely Filter, Draw, and Bundle. We then propose two alternative implementations and experimentally compare them against the state-of-the-art approach and the simple idea of drawing and subsequently bundling the graph. The experiments confirm that bundled drawings created by our framework outperform previous approaches according to standard quality metrics for edge bundling. © Daniel Archambault, Giuseppe Liotta, Martin Nöllenburg, Tommaso Piselli, Alessandra Tappini, and Markus Wallinger.

关键词： graph algorithms

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Intersection graphs with and Without Product Structure 32

Intersection Graphs with and Without Product Structure

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32nd international symposium on graph drawing and network visualization, gd 2024

作者： Merker, Laura Scherzer, Lena Schneider, Samuel Ueckerdt, Torsten Germany

ISBN: (纸本)9783959773430

A graph class G admits product structure if there exists a constant k such that every G ∈ G is a subgraph of H ⊠ P for a path P and some graph H of treewidth k. Famously, the class of planar graphs, as well as many beyond-planar graph classes are known to admit product structure. However, we have only few tools to prove the absence of product structure, and hence know of only a few interesting examples of classes. Motivated by the transition between product structure and no product structure, we investigate subclasses of intersection graphs in the plane (e.g., disk intersection graphs) and present necessary and sufficient conditions for these to admit product structure. Specifically, for a set S ⊂ R2 (e.g., a disk) and a real number α ∈ [0, 1], we consider intersection graphs of α-free homothetic copies of S. That is, each vertex v is a homothetic copy of S of which at least an α-portion is not covered by other vertices, and there is an edge between u and v if and only if u ∩ v ≠ ∅. For α = 1 we have contact graphs, which are in most cases planar, and hence admit product structure. For α = 0 we have (among others) all complete graphs, and hence no product structure. In general, there is a threshold value α∗(S) ∈ [0, 1] such that α-free homothetic copies of S admit product structure for all α > α∗(S) and do not admit product structure for all α © Laura Merker, Lena Scherzer, Samuel Schneider, and Torsten Ueckerdt.

关键词： graph structures

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