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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} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

A map graph is one admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. The algorithm has time complexity that is linear in the size of the graph and, if the input is a yes-instance, it reports a certificate in the form of a so-called witness. Furthermore, this result is developed within a more general algorithmic framework that allows to test, for any k, if the input graph admits a k-map (where at most k nations meet at a common point) or a hole-free k-map (where each point is covered by at least one nation). We point out that, although bounding the treewidth of the input graph also bounds the size of its largest clique, the latter alone does not seem to be a strong enough structural limitation to obtain an efficient time complexity. In fact, while the largest clique in a k-map graph is ⌊ 3k/2 ⌋, the recognition of k-map graphs is still open for any fixed k ≥ 5.

Patrizio Angelini, Michael A. Bekos, Giordano Da Lozzo, Martin Gronemann, Fabrizio Montecchiani, and Alessandra Tappini. Recognizing Map Graphs of Bounded Treewidth. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{angelini_et_al:LIPIcs.SWAT.2022.8, author = {Angelini, Patrizio and Bekos, Michael A. and Da Lozzo, Giordano and Gronemann, Martin and Montecchiani, Fabrizio and Tappini, Alessandra}, title = {{Recognizing Map Graphs of Bounded Treewidth}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.8}, URN = {urn:nbn:de:0030-drops-161681}, doi = {10.4230/LIPIcs.SWAT.2022.8}, annote = {Keywords: Map graphs, Recognition, Parameterized complexity} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs.

Michael A. Bekos, Giordano Da Lozzo, Svenja M. Griesbach, Martin Gronemann, Fabrizio Montecchiani, and Chrysanthi Raftopoulou. Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bekos_et_al:LIPIcs.SoCG.2020.16, author = {Bekos, Michael A. and Da Lozzo, Giordano and Griesbach, Svenja M. and Gronemann, Martin and Montecchiani, Fabrizio and Raftopoulou, Chrysanthi}, title = {{Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {16:1--16:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.16}, URN = {urn:nbn:de:0030-drops-121749}, doi = {10.4230/LIPIcs.SoCG.2020.16}, annote = {Keywords: Book embeddings, Book thickness, Nonplanar graphs, Planar skeleton} }

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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)

We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).

Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, and Fabrizio Montecchiani. A Universal Slope Set for 1-Bend Planar Drawings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{angelini_et_al:LIPIcs.SoCG.2017.9, author = {Angelini, Patrizio and Bekos, Michael A. and Liotta, Giuseppe and Montecchiani, Fabrizio}, title = {{A Universal Slope Set for 1-Bend Planar Drawings}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {9:1--9:16}, 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.2017.9}, URN = {urn:nbn:de:0030-drops-71917}, doi = {10.4230/LIPIcs.SoCG.2017.9}, annote = {Keywords: Slope number, 1-bend drawings, planar graphs, angular resolution} }

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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:** 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:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Back in the eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book embedding viewpoint of the problem.

Michael A. Bekos, Martin Gronemann, and Chrysanthi N. Raftopoulou. Two-Page Book Embeddings of 4-Planar Graphs. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 137-148, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bekos_et_al:LIPIcs.STACS.2014.137, author = {Bekos, Michael A. and Gronemann, Martin and Raftopoulou, Chrysanthi N.}, title = {{Two-Page Book Embeddings of 4-Planar Graphs}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {137--148}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.137}, URN = {urn:nbn:de:0030-drops-44536}, doi = {10.4230/LIPIcs.STACS.2014.137}, annote = {Keywords: Book Embedding, Subhamiltonicity, 4-Planar Graphs} }

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