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Documents authored by Dujmovic, Vida

Found 2 Possible Name Variants:

Dujmovic, Vida

Document
DOI: 10.4230/LIPIcs.GD.2024.27
On k-Planar Graphs Without Short Cycles

Authors: Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c√kn, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.
Cite as

Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger. On k-Planar Graphs Without Short Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bekos_et_al:LIPIcs.GD.2024.27,
  author =	{Bekos, Michael A. and Bose, Prosenjit and B\"{u}ngener, Aaron and Dujmovi\'{c}, Vida and Hoffmann, Michael and Kaufmann, Michael and Morin, Pat and Odak, Saeed and Weinberger, Alexandra},
  title =	{{On k-Planar Graphs Without Short Cycles}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.27},
  URN =		{urn:nbn:de:0030-drops-213117},
  doi =		{10.4230/LIPIcs.GD.2024.27},
  annote =	{Keywords: Beyond-planar Graphs, k-planar Graphs, Local Crossing Number, Crossing Number, Discharging Method, Crossing Lemma}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.37
Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor

Authors: Vida Dujmović and Camille La Rose

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
The rectilinear crossing number of G is the minimum number of crossings in a straight-line drawing of G. A single-crossing graph is a graph whose crossing number is at most one. We prove that every n-vertex graph G that excludes a single-crossing graph as a minor has rectilinear crossing number O(Δ n), where Δ is the maximum degree of G. This dependence on n and Δ is best possible. The result applies, for example, to K₅-minor-free graphs, and bounded treewidth graphs. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth as well as bounded degree K_{3,3}-minor-free graphs. In the case of bounded treewidth graphs, our O(Δ n) result is again tight and it improves on the previous best known bound of O(Δ² n) by Wood and Telle, 2007.
Cite as

Vida Dujmović and Camille La Rose. Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dujmovic_et_al:LIPIcs.GD.2024.37,
  author =	{Dujmovi\'{c}, Vida and La Rose, Camille},
  title =	{{Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.37},
  URN =		{urn:nbn:de:0030-drops-213219},
  doi =		{10.4230/LIPIcs.GD.2024.37},
  annote =	{Keywords: (rectilinear) crossing number, graph minors, maximum degree, clique-sums}
}
Document
DOI: 10.4230/DagRep.14.2.71
Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062)

Authors: Vida Dujmović, Seok-Hee Hong, Michael Kaufmann, János Pach, and Henry Förster

Published in: Dagstuhl Reports, Volume 14, Issue 2 (2024)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 24062 "Beyond-Planar Graphs: Models, Structures and Geometric Representations". The seminar investigated beyond-planar graphs, in particular, their combinatorial and topological structures, computational complexity and algorithmics for recognition, geometric representations, and their applications to real-world network visualization. Compared to the previous two editions of the seminar, we focus more on aspects of combinatorics and geometry. The program consists of four invited talks on beyond planar graphs, open problem session, problem solving sessions and progress report sessions. Specific open problems include questions regarding the combinatorial structures and topology (e.g., k^+-real face graphs, beyond upward planar graphs, sparse universal geometric graphs, local-crossing-critical graphs), the geometric representations (e.g., constrained outer string graphs, rerouting curves on surface), and applications. The details of the invited talks and progress reports from each working groups are included in this report.
Cite as

Vida Dujmović, Seok-Hee Hong, Michael Kaufmann, János Pach, and Henry Förster. Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062). In Dagstuhl Reports, Volume 14, Issue 2, pp. 71-94, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{dujmovic_et_al:DagRep.14.2.71,
  author =	{Dujmovi\'{c}, Vida and Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and F\"{o}rster, Henry},
  title =	{{Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062)}},
  pages =	{71--94},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2024},
  volume =	{14},
  number =	{2},
  editor =	{Dujmovi\'{c}, Vida and Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and F\"{o}rster, Henry},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.2.71},
  URN =		{urn:nbn:de:0030-drops-204999},
  doi =		{10.4230/DagRep.14.2.71},
  annote =	{Keywords: Combinatorial geometry, Graph algorithm, Graph drawing, Graph theory, Network visualization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.3
Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers

Authors: Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada, André van Renssen, and Vera Sacristán

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra "helper" modules ("musketeers") suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses O(n^2) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive "sliding" moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.
Cite as

Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada, André van Renssen, and Vera Sacristán. Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akitaya_et_al:LIPIcs.ESA.2019.3,
  author =	{Akitaya, Hugo A. and Arkin, Esther M. and Damian, Mirela and Demaine, Erik D. and Dujmovi\'{c}, Vida and Flatland, Robin and Korman, Matias and Palop, Belen and Parada, Irene and van Renssen, Andr\'{e} and Sacrist\'{a}n, Vera},
  title =	{{Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.3},
  URN =		{urn:nbn:de:0030-drops-111247},
  doi =		{10.4230/LIPIcs.ESA.2019.3},
  annote =	{Keywords: Reconfiguration, geometric algorithm, pivoting squares, modular robots}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.29
Dual Circumference and Collinear Sets

Authors: Vida Dujmović and Pat Morin

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
We show that, if an n-vertex triangulation T of maximum degree Delta has a dual that contains a cycle of length l, then T has a non-crossing straight-line drawing in which some set, called a collinear set, of Omega(l/Delta^4) vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every n-vertex planar graph of maximum degree Delta has a collinear set of size Omega(n^{0.8}/Delta^4). Very recently, Dujmović et al. (SODA 2019) showed that, if S is a collinear set in a triangulation T then, for any point set X subset R^2 with |X|=|S|, T has a non-crossing straight-line drawing in which the vertices of S are drawn on the points in X. Because of this, collinear sets have numerous applications in graph drawing and related areas.
Cite as

Vida Dujmović and Pat Morin. Dual Circumference and Collinear Sets. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dujmovic_et_al:LIPIcs.SoCG.2019.29,
  author =	{Dujmovi\'{c}, Vida and Morin, Pat},
  title =	{{Dual Circumference and Collinear Sets}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.29},
  URN =		{urn:nbn:de:0030-drops-104338},
  doi =		{10.4230/LIPIcs.SoCG.2019.29},
  annote =	{Keywords: Planar graphs, collinear sets, untangling, column planarity, universal point subsets, partial simultaneous geometric drawings}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.23
Geodesic Obstacle Representation of Graphs

Authors: Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles. In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.
Cite as

Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira. Geodesic Obstacle Representation of Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.ICALP.2018.23,
  author =	{Bose, Prosenjit and Carmi, Paz and Dujmovic, Vida and Mehrabi, Saeed and Montecchiani, Fabrizio and Morin, Pat and Silveira, Luis Fernando Schultz Xavier da},
  title =	{{Geodesic Obstacle Representation of Graphs}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{23:1--23:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.23},
  URN =		{urn:nbn:de:0030-drops-90274},
  doi =		{10.4230/LIPIcs.ICALP.2018.23},
  annote =	{Keywords: Obstacle representation, Grid obstacle representation, Geodesic obstacle representation}
}

Dujmović, Vida

Document
DOI: 10.4230/LIPIcs.GD.2024.27
On k-Planar Graphs Without Short Cycles

Authors: Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c√kn, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.
Cite as

Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger. On k-Planar Graphs Without Short Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bekos_et_al:LIPIcs.GD.2024.27,
  author =	{Bekos, Michael A. and Bose, Prosenjit and B\"{u}ngener, Aaron and Dujmovi\'{c}, Vida and Hoffmann, Michael and Kaufmann, Michael and Morin, Pat and Odak, Saeed and Weinberger, Alexandra},
  title =	{{On k-Planar Graphs Without Short Cycles}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.27},
  URN =		{urn:nbn:de:0030-drops-213117},
  doi =		{10.4230/LIPIcs.GD.2024.27},
  annote =	{Keywords: Beyond-planar Graphs, k-planar Graphs, Local Crossing Number, Crossing Number, Discharging Method, Crossing Lemma}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.37
Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor

Authors: Vida Dujmović and Camille La Rose

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
The rectilinear crossing number of G is the minimum number of crossings in a straight-line drawing of G. A single-crossing graph is a graph whose crossing number is at most one. We prove that every n-vertex graph G that excludes a single-crossing graph as a minor has rectilinear crossing number O(Δ n), where Δ is the maximum degree of G. This dependence on n and Δ is best possible. The result applies, for example, to K₅-minor-free graphs, and bounded treewidth graphs. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth as well as bounded degree K_{3,3}-minor-free graphs. In the case of bounded treewidth graphs, our O(Δ n) result is again tight and it improves on the previous best known bound of O(Δ² n) by Wood and Telle, 2007.
Cite as

Vida Dujmović and Camille La Rose. Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dujmovic_et_al:LIPIcs.GD.2024.37,
  author =	{Dujmovi\'{c}, Vida and La Rose, Camille},
  title =	{{Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.37},
  URN =		{urn:nbn:de:0030-drops-213219},
  doi =		{10.4230/LIPIcs.GD.2024.37},
  annote =	{Keywords: (rectilinear) crossing number, graph minors, maximum degree, clique-sums}
}
Document
DOI: 10.4230/DagRep.14.2.71
Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062)

Authors: Vida Dujmović, Seok-Hee Hong, Michael Kaufmann, János Pach, and Henry Förster

Published in: Dagstuhl Reports, Volume 14, Issue 2 (2024)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 24062 "Beyond-Planar Graphs: Models, Structures and Geometric Representations". The seminar investigated beyond-planar graphs, in particular, their combinatorial and topological structures, computational complexity and algorithmics for recognition, geometric representations, and their applications to real-world network visualization. Compared to the previous two editions of the seminar, we focus more on aspects of combinatorics and geometry. The program consists of four invited talks on beyond planar graphs, open problem session, problem solving sessions and progress report sessions. Specific open problems include questions regarding the combinatorial structures and topology (e.g., k^+-real face graphs, beyond upward planar graphs, sparse universal geometric graphs, local-crossing-critical graphs), the geometric representations (e.g., constrained outer string graphs, rerouting curves on surface), and applications. The details of the invited talks and progress reports from each working groups are included in this report.
Cite as

Vida Dujmović, Seok-Hee Hong, Michael Kaufmann, János Pach, and Henry Förster. Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062). In Dagstuhl Reports, Volume 14, Issue 2, pp. 71-94, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{dujmovic_et_al:DagRep.14.2.71,
  author =	{Dujmovi\'{c}, Vida and Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and F\"{o}rster, Henry},
  title =	{{Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062)}},
  pages =	{71--94},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2024},
  volume =	{14},
  number =	{2},
  editor =	{Dujmovi\'{c}, Vida and Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and F\"{o}rster, Henry},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.2.71},
  URN =		{urn:nbn:de:0030-drops-204999},
  doi =		{10.4230/DagRep.14.2.71},
  annote =	{Keywords: Combinatorial geometry, Graph algorithm, Graph drawing, Graph theory, Network visualization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.3
Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers

Authors: Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada, André van Renssen, and Vera Sacristán

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra "helper" modules ("musketeers") suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses O(n^2) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive "sliding" moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.
Cite as

Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada, André van Renssen, and Vera Sacristán. Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akitaya_et_al:LIPIcs.ESA.2019.3,
  author =	{Akitaya, Hugo A. and Arkin, Esther M. and Damian, Mirela and Demaine, Erik D. and Dujmovi\'{c}, Vida and Flatland, Robin and Korman, Matias and Palop, Belen and Parada, Irene and van Renssen, Andr\'{e} and Sacrist\'{a}n, Vera},
  title =	{{Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.3},
  URN =		{urn:nbn:de:0030-drops-111247},
  doi =		{10.4230/LIPIcs.ESA.2019.3},
  annote =	{Keywords: Reconfiguration, geometric algorithm, pivoting squares, modular robots}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.29
Dual Circumference and Collinear Sets

Authors: Vida Dujmović and Pat Morin

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
We show that, if an n-vertex triangulation T of maximum degree Delta has a dual that contains a cycle of length l, then T has a non-crossing straight-line drawing in which some set, called a collinear set, of Omega(l/Delta^4) vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every n-vertex planar graph of maximum degree Delta has a collinear set of size Omega(n^{0.8}/Delta^4). Very recently, Dujmović et al. (SODA 2019) showed that, if S is a collinear set in a triangulation T then, for any point set X subset R^2 with |X|=|S|, T has a non-crossing straight-line drawing in which the vertices of S are drawn on the points in X. Because of this, collinear sets have numerous applications in graph drawing and related areas.
Cite as

Vida Dujmović and Pat Morin. Dual Circumference and Collinear Sets. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dujmovic_et_al:LIPIcs.SoCG.2019.29,
  author =	{Dujmovi\'{c}, Vida and Morin, Pat},
  title =	{{Dual Circumference and Collinear Sets}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.29},
  URN =		{urn:nbn:de:0030-drops-104338},
  doi =		{10.4230/LIPIcs.SoCG.2019.29},
  annote =	{Keywords: Planar graphs, collinear sets, untangling, column planarity, universal point subsets, partial simultaneous geometric drawings}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.23
Geodesic Obstacle Representation of Graphs

Authors: Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles. In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.
Cite as

Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira. Geodesic Obstacle Representation of Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.ICALP.2018.23,
  author =	{Bose, Prosenjit and Carmi, Paz and Dujmovic, Vida and Mehrabi, Saeed and Montecchiani, Fabrizio and Morin, Pat and Silveira, Luis Fernando Schultz Xavier da},
  title =	{{Geodesic Obstacle Representation of Graphs}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{23:1--23:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.23},
  URN =		{urn:nbn:de:0030-drops-90274},
  doi =		{10.4230/LIPIcs.ICALP.2018.23},
  annote =	{Keywords: Obstacle representation, Grid obstacle representation, Geodesic obstacle representation}
}
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