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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number.
We show that the page number of an ordered graph with n vertices and m edges can be computed in 2^m ⋅ n^𝒪(1) time. An 𝒪(log n)-approximation of this number can be computed efficiently. We can decide in 2^𝒪(d √k log (d+k)) ⋅ n^𝒪(1) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h = 1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h > 1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in 2ⁿ ⋅ n^𝒪(1) time either the number of crossings or, if we disallow crossings, the number of tracks.

Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating Crossings in Ordered Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2024.1, author = {Agrawal, Akanksha and Cabello, Sergio and Kaufmann, Michael and Saurabh, Saket and Sharma, Roohani and Uno, Yushi and Wolff, Alexander}, title = {{Eliminating Crossings in Ordered Graphs}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.1}, URN = {urn:nbn:de:0030-drops-200417}, doi = {10.4230/LIPIcs.SWAT.2024.1}, annote = {Keywords: Ordered graphs, book embedding, edge deletion, d-planar, hitting set} }

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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

A rectilinear-upward planar drawing of a digraph G is a crossing-free drawing of G where each edge is either a horizontal or a vertical segment, and such that no directed edge points downward. Rectilinear-Upward Planarity Testing is the problem of deciding whether a digraph G admits a rectilinear-upward planar drawing. We show that: (i) Rectilinear-Upward Planarity Testing is NP-complete, even if G is biconnected; (ii) it can be solved in linear time when an upward planar embedding of G is fixed; (iii) the problem is polynomial-time solvable for biconnected digraphs of treewidth at most two, i.e., for digraphs whose underlying undirected graph is a series-parallel graph; (iv) for any biconnected digraph the problem is fixed-parameter tractable when parameterized by the number of sources and sinks in the digraph.

Walter Didimo, Michael Kaufmann, Giuseppe Liotta, Giacomo Ortali, and Maurizio Patrignani. Rectilinear-Upward Planarity Testing of Digraphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 26:1-26:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{didimo_et_al:LIPIcs.ISAAC.2023.26, author = {Didimo, Walter and Kaufmann, Michael and Liotta, Giuseppe and Ortali, Giacomo and Patrignani, Maurizio}, title = {{Rectilinear-Upward Planarity Testing of Digraphs}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {26:1--26:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.26}, URN = {urn:nbn:de:0030-drops-193283}, doi = {10.4230/LIPIcs.ISAAC.2023.26}, annote = {Keywords: Graph drawing, orthogonal drawings, upward drawings, rectilinear planarity, upward planarity} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity.
In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model.

Patrizio Angelini, Michael A. Bekos, Julia Katheder, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt. Axis-Parallel Right Angle Crossing Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{angelini_et_al:LIPIcs.ESA.2023.9, author = {Angelini, Patrizio and Bekos, Michael A. and Katheder, Julia and Kaufmann, Michael and Pfister, Maximilian and Ueckerdt, Torsten}, title = {{Axis-Parallel Right Angle Crossing Graphs}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {9:1--9:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.9}, URN = {urn:nbn:de:0030-drops-186623}, doi = {10.4230/LIPIcs.ESA.2023.9}, annote = {Keywords: Graph drawing, RAC graphs, Graph drawing algorithms} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmović et al., J. ACM, 67(4), 22:1-38, 2020].
In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3⌊ h/2 ⌋+⌊ h/3 ⌋-1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 115 to 82 and from ⌊ 33/2(k+3 ⌊ k/2⌋ -3)⌋ to ⌊ 33/2 (3⌊ k/2 ⌋+⌊ k/3 ⌋-1) ⌋, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of 1-planar graphs from O(1) to 80. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.

Michael A. Bekos, Giordano Da Lozzo, Petr Hliněný, and Michael Kaufmann. Graph Product Structure for h-Framed Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bekos_et_al:LIPIcs.ISAAC.2022.23, author = {Bekos, Michael A. and Da Lozzo, Giordano and Hlin\v{e}n\'{y}, Petr and Kaufmann, Michael}, title = {{Graph Product Structure for h-Framed Graphs}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.23}, URN = {urn:nbn:de:0030-drops-173086}, doi = {10.4230/LIPIcs.ISAAC.2022.23}, annote = {Keywords: Graph product structure theory, h-framed graphs, k-map graphs, queue number, twin-width} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90° angles.
In this work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hamiltonian degree-3 graphs. We improve on this result by extending to the class of 3-edge-colorable degree-3 graphs. When each edge is allowed to have one bend, we prove that degree-4 graphs admit such RAC drawings, a result which was previously known only for degree-3 graphs. Finally, we show that 7-edge-colorable degree-7 graphs admit RAC drawings with two bends per edge. This improves over the previous result on degree-6 graphs.

Patrizio Angelini, Michael A. Bekos, Julia Katheder, Michael Kaufmann, and Maximilian Pfister. RAC Drawings of Graphs with Low Degree. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{angelini_et_al:LIPIcs.MFCS.2022.11, author = {Angelini, Patrizio and Bekos, Michael A. and Katheder, Julia and Kaufmann, Michael and Pfister, Maximilian}, title = {{RAC Drawings of Graphs with Low Degree}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {11:1--11:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.11}, URN = {urn:nbn:de:0030-drops-168090}, doi = {10.4230/LIPIcs.MFCS.2022.11}, annote = {Keywords: Graph Drawing, RAC graphs, Straight-line and bent drawings} }

Document

**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We present new bounds for the required area of Right Angle Crossing (RAC) drawings for complete graphs, i.e. drawings where any two crossing edges are perpendicular to each other. First, we improve upon results by Didimo et al. [Walter Didimo et al., 2011] and Di Giacomo et al. [Emilio Di Giacomo et al., 2011] by showing how to compute a RAC drawing with three bends per edge in cubic area. We also show that quadratic area can be achieved when allowing eight bends per edge in general or with three bends per edge for p-partite graphs. As a counterpart, we prove that in general quadratic area is not sufficient for RAC drawings with three bends per edge.

Henry Förster and Michael Kaufmann. On Compact RAC Drawings. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{forster_et_al:LIPIcs.ESA.2020.53, author = {F\"{o}rster, Henry and Kaufmann, Michael}, title = {{On Compact RAC Drawings}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {53:1--53:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.53}, URN = {urn:nbn:de:0030-drops-129192}, doi = {10.4230/LIPIcs.ESA.2020.53}, annote = {Keywords: RAC drawings, visualization of dense graphs, compact drawings} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. In this paper, we study fan-planar drawings that use h (horizontal) layers and are proper, i.e., edges connect adjacent layers. We show that if the embedding of the graph is fixed, then testing the existence of such drawings is fixed-parameter tractable in h, via a reduction to a similar result for planar graphs by Dujmović et al. If the embedding is not fixed, then we give partial results for h = 2: It was already known how to test the existence of fan-planar proper 2-layer drawings for 2-connected graphs, and we show here how to test this for trees. Along the way, we exhibit other interesting results for graphs with a fan-planar proper h-layer drawing; in particular we bound their pathwidth and show that they have a bar-1-visibility representation.

Therese Biedl, Steven Chaplick, Michael Kaufmann, Fabrizio Montecchiani, Martin Nöllenburg, and Chrysanthi Raftopoulou. Layered Fan-Planar Graph Drawings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{biedl_et_al:LIPIcs.MFCS.2020.14, author = {Biedl, Therese and Chaplick, Steven and Kaufmann, Michael and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Raftopoulou, Chrysanthi}, title = {{Layered Fan-Planar Graph Drawings}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.14}, URN = {urn:nbn:de:0030-drops-126835}, doi = {10.4230/LIPIcs.MFCS.2020.14}, annote = {Keywords: Graph Drawing, Parameterized Complexity, Beyond Planar Graphs} }

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**Published in:** Dagstuhl Reports, Volume 9, Issue 2 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory, combinatorics, computational geometry, and graph
drawing. This seminar continued the work initiated in Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics" and focused on the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs that admit a drawing with topological constraints such as specific types of crossings, or with some forbidden crossing patterns.
The seminar began with four talks about the results of scientific collaborations originating from the previous Dagstuhl seminar. Next we discussed open research problems
about beyond planar graphs, such as their combinatorial structures (e.g., book thickness, queue number), their topology (e.g., simultaneous embeddability, gap planarity, quasi-quasiplanarity), their geometric representations
(e.g., representations on few segments or arcs), and applications
(e.g., manipulation of graph drawings by untangling operations). Six working groups were formed that investigated several of the open research questions. In addition, talks on related subjects and recent conference contributions were presented in the morning opening sessions. Abstracts of all talks and a report from each working group are included in this report.

Seok-Hee Hong, Michael Kaufmann, János Pach, and Csaba D. Tóth. Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092). In Dagstuhl Reports, Volume 9, Issue 2, pp. 123-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{hong_et_al:DagRep.9.2.123, author = {Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.}, title = {{Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092)}}, pages = {123--156}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {2}, editor = {Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.2.123}, URN = {urn:nbn:de:0030-drops-108634}, doi = {10.4230/DagRep.9.2.123}, annote = {Keywords: combinatorial geometry, geometric algorithms, graph algorithms, graph drawing, graph theory, network visualization} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter due to edge crossings should be minimized (e.g., Huang et al. 2018).
Most of the literature focuses on Turán-type problems, which ask for the maximum number of edges a beyond-planar graph can have. Here, we study this problem for bipartite topological graphs, considering several types of beyond-planar graphs, i.e. 1-planar, 2-planar, fan-planar, and RAC graphs. We prove bounds on the number of edges that are tight up to additive constants; some of them are surprising and not along the lines of the known results for non-bipartite graphs. Our findings lead to an improvement of the leading constant of the well-known Crossing Lemma for bipartite graphs, as well as to a number of interesting questions on topological graphs.

Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt. Beyond-Planarity: Turán-Type Results for Non-Planar Bipartite Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{angelini_et_al:LIPIcs.ISAAC.2018.28, author = {Angelini, Patrizio and Bekos, Michael A. and Kaufmann, Michael and Pfister, Maximilian and Ueckerdt, Torsten}, title = {{Beyond-Planarity: Tur\'{a}n-Type Results for Non-Planar Bipartite Graphs}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.28}, URN = {urn:nbn:de:0030-drops-99763}, doi = {10.4230/LIPIcs.ISAAC.2018.28}, annote = {Keywords: Bipartite topological graphs, beyond planarity, density, Crossing Lemma} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

A graph is k-planar if it can be drawn in the plane such that no edge is crossed more than k times. While for k=1, optimal 1-planar graphs, i.e., those with n vertices and exactly 4n-8 edges, have been completely characterized, this has not been the case for k > 1. For k=2,3 and 4, upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple 2- and 3-planar graphs without homotopic parallel edges and self-loops.
In this paper, we completely characterize optimal 2- and 3-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.

Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. On Optimal 2- and 3-Planar Graphs. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bekos_et_al:LIPIcs.SoCG.2017.16, author = {Bekos, Michael A. and Kaufmann, Michael and Raftopoulou, Chrysanthi N.}, title = {{On Optimal 2- and 3-Planar Graphs}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.16}, URN = {urn:nbn:de:0030-drops-72307}, doi = {10.4230/LIPIcs.SoCG.2017.16}, annote = {Keywords: topological graphs, optimal k-planar graphs, characterization} }

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**Published in:** Dagstuhl Reports, Volume 6, Issue 11 (2017)

This report summarizes Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics'' and documents the talks and discussions.
The seminar brought together 29 researchers in the areas of graph theory, combinatorics, computational geometry, and graph drawing. The common interest was in the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs with topological constraints such as specific types of crossings, or with some forbidden crossing patterns. The seminar began with three introductory talks by experts in the different fields. Abstracts of these talks are collected in this report. Next we discussed and grouped together open research problems about beyond planar graphs, such as their combinatorial structures (e.g, thickness, crossing number, coloring), their topology (e.g., string graph representation), their geometric representations (e.g., straight-line drawing, visibility representation, contact representation), and applications (e.g., algorithms for real-world network visualization). Four working groups were formed and a report from each group is included here.

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, and János Pach. Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452). In Dagstuhl Reports, Volume 6, Issue 11, pp. 35-62, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{hong_et_al:DagRep.6.11.35, author = {Hong, Sok-Hee and Kaufmann, Michael and Kobourov, Stephen G. and Pach, J\'{a}nos}, title = {{Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)}}, pages = {35--62}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {6}, number = {11}, editor = {Hong, Sok-Hee and Kaufmann, Michael and Kobourov, Stephen G. and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.6.11.35}, URN = {urn:nbn:de:0030-drops-70385}, doi = {10.4230/DagRep.6.11.35}, annote = {Keywords: graph drawing, graph algorithms, graph theory, geometric algorithms, combinatorial geometry, visualization} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garcia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.

Markus Geyer, Michael Hoffmann, Michael Kaufmann, Vincent Kusters, and Csaba Tóth. The Planar Tree Packing Theorem. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{geyer_et_al:LIPIcs.SoCG.2016.41, author = {Geyer, Markus and Hoffmann, Michael and Kaufmann, Michael and Kusters, Vincent and T\'{o}th, Csaba}, title = {{The Planar Tree Packing Theorem}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {41:1--41:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.41}, URN = {urn:nbn:de:0030-drops-59337}, doi = {10.4230/LIPIcs.SoCG.2016.41}, annote = {Keywords: graph drawing, simultaneous embedding, planar graph, graph packin} }

Document

**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

We discuss algorithmic issues on the well-known paper-and-pencil game RaceTrack. On a very simple track called Indianapolis, we introduce the problem and simple approaches, that will be gradually refined. We present and experimentally evaluate efficient algorithms for single player scenarios. We also consider a variant where the parts of the track are known as soon as they become visible during the race.

Michael A. Bekos, Till Bruckdorfer, Henry Förster, Michael Kaufmann, Simon Poschenrieder, and Thomas Stüber. Algorithms and Insights for RaceTrack. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bekos_et_al:LIPIcs.FUN.2016.6, author = {Bekos, Michael A. and Bruckdorfer, Till and F\"{o}rster, Henry and Kaufmann, Michael and Poschenrieder, Simon and St\"{u}ber, Thomas}, title = {{Algorithms and Insights for RaceTrack}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.6}, URN = {urn:nbn:de:0030-drops-58818}, doi = {10.4230/LIPIcs.FUN.2016.6}, annote = {Keywords: Racetrack, State-graph, complexity} }

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**Published in:** Dagstuhl Reports, Volume 1, Issue 5 (2011)

This report documents the program and the outcomes of Dagstuhl Seminar 11191
``Graph Drawing with Algorithm Engineering Methods''. We summarize the talks,
open problems, and working group discussions.

Camil Demetrescu, Michael Kaufmann, Stephen Kobourov, and Petra Mutzel. Graph Drawing with Algorithm Engineering Methods (Dagstuhl Seminar 11191). In Dagstuhl Reports, Volume 1, Issue 5, pp. 47-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@Article{demetrescu_et_al:DagRep.1.5.47, author = {Demetrescu, Camil and Kaufmann, Michael and Kobourov, Stephen and Mutzel, Petra}, title = {{Graph Drawing with Algorithm Engineering Methods (Dagstuhl Seminar 11191)}}, pages = {47--60}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2011}, volume = {1}, number = {5}, editor = {Demetrescu, Camil and Kaufmann, Michael and Kobourov, Stephen and Mutzel, Petra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.5.47}, URN = {urn:nbn:de:0030-drops-32046}, doi = {10.4230/DagRep.1.5.47}, annote = {Keywords: Algorithm Engineering, Graph Drawing} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8191, Graph Drawing with Applications to Bioinformatics and Social Sciences (2008)

We considered the following problem: Given a set of vertices V and a set of paths
P, where each path is a sequence of vertices, represent these paths somehow.
We explored representations in different dimensions and with different conditions on the paths.

Stephen Borgatti, Ulrik Brandes, Michael Kaufmann, Stephen Kobourov, Anna Lubiw, and Dorothea Wagner. 08191 Working Group Report – Visualization of Trajectories. In Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl Seminar Proceedings, Volume 8191, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{borgatti_et_al:DagSemProc.08191.4, author = {Borgatti, Stephen and Brandes, Ulrik and Kaufmann, Michael and Kobourov, Stephen and Lubiw, Anna and Wagner, Dorothea}, title = {{08191 Working Group Report – Visualization of Trajectories}}, booktitle = {Graph Drawing with Applications to Bioinformatics and Social Sciences}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8191}, editor = {Stephen P. Borgatti and Stephen Kobourov and Oliver Kohlbacher and Petra Mutzel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08191.4}, URN = {urn:nbn:de:0030-drops-15558}, doi = {10.4230/DagSemProc.08191.4}, annote = {Keywords: Graph drawing, trajectories, paths} }