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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph K_n admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length Ω(log n / log log n) and of a plane matching of size Ω(n^{1/2}) in every simple drawing of K_n.
Instead of studying simpler substructures, we prove Rafla’s conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of K_n contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane k-cycle for each 3 ≤ k ≤ n (pancyclicity), and present further results on maximal plane subdrawings.

Helena Bergold, Stefan Felsner, Meghana M. Reddy, Joachim Orthaber, and Manfred Scheucher. Plane Hamiltonian Cycles in Convex Drawings. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bergold_et_al:LIPIcs.SoCG.2024.18, author = {Bergold, Helena and Felsner, Stefan and M. Reddy, Meghana and Orthaber, Joachim and Scheucher, Manfred}, title = {{Plane Hamiltonian Cycles in Convex Drawings}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.18}, URN = {urn:nbn:de:0030-drops-199630}, doi = {10.4230/LIPIcs.SoCG.2024.18}, annote = {Keywords: simple drawing, convexity hierarchy, plane pancyclicity, plane Hamiltonian connectivity, maximal plane subdrawing} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Arrangements of pseudolines are classic objects in discrete and computational geometry. They have been studied with increasing intensity since their introduction almost 100 years ago. The study of the number B_n of non-isomorphic simple arrangements of n pseudolines goes back to Goodman and Pollack, Knuth, and others. It is known that B_n is in the order of 2^Θ(n²) and finding asymptotic bounds on b_n = log₂(B_n)/n² remains a challenging task. In 2011, Felsner and Valtr showed that 0.1887 ≤ b_n ≤ 0.6571 for sufficiently large n. The upper bound remains untouched but in 2020 Dumitrescu and Mandal improved the lower bound constant to 0.2083. Their approach utilizes the known values of B_n for up to n = 12.
We tackle the lower bound by utilizing dynamic programming and the Lindström–Gessel–Viennot lemma. Our new bound is b_n ≥ 0.2721 for sufficiently large n. The result is based on a delicate interplay of theoretical ideas and computer assistance.

Fernando Cortés Kühnast, Justin Dallant, Stefan Felsner, and Manfred Scheucher. An Improved Lower Bound on the Number of Pseudoline Arrangements. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{corteskuhnast_et_al:LIPIcs.SoCG.2024.43, author = {Cort\'{e}s K\"{u}hnast, Fernando and Dallant, Justin and Felsner, Stefan and Scheucher, Manfred}, title = {{An Improved Lower Bound on the Number of Pseudoline Arrangements}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.43}, URN = {urn:nbn:de:0030-drops-199880}, doi = {10.4230/LIPIcs.SoCG.2024.43}, annote = {Keywords: counting, pseudoline arrangement, recursive construction, bipermutation, divide and conquer, dynamic programming, computer-assisted proof} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In 1926, Levi showed that, for every pseudoline arrangement 𝒜 and two points in the plane, 𝒜 can be extended by a pseudoline which contains the two prescribed points. Later extendability was studied for arrangements of pseudohyperplanes in higher dimensions. While the extendability of an arrangement of proper hyperplanes in ℝ^d with a hyperplane containing d prescribed points is trivial, Richter-Gebert found an arrangement of pseudoplanes in ℝ³ which cannot be extended with a pseudoplane containing two particular prescribed points.
In this article, we investigate the extendability of signotopes, which are a combinatorial structure encoding a rich subclass of pseudohyperplane arrangements. Our main result is that signotopes of odd rank are extendable in the sense that for two prescribed crossing points we can add an element containing them. Moreover, we conjecture that in all even ranks r ≥ 4 there exist signotopes which are not extendable for two prescribed points. Our conjecture is supported by examples in ranks 4, 6, 8, 10, and 12 that were found with a SAT based approach.

Helena Bergold, Stefan Felsner, and Manfred Scheucher. An Extension Theorem for Signotopes. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergold_et_al:LIPIcs.SoCG.2023.17, author = {Bergold, Helena and Felsner, Stefan and Scheucher, Manfred}, title = {{An Extension Theorem for Signotopes}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {17:1--17:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.17}, URN = {urn:nbn:de:0030-drops-178676}, doi = {10.4230/LIPIcs.SoCG.2023.17}, annote = {Keywords: arrangement of pseudolines, extendability, Levi’s extension lemma, arrangement of pseudohyperplanes, signotope, oriented matroid, partial order, Boolean satisfiability (SAT)} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

A finite set P of points in the plane is n-universal with respect to a class 𝒞 of planar graphs if every n-vertex graph in 𝒞 admits a crossing-free straight-line drawing with vertices at points of P.
For the class of all planar graphs the best known upper bound on the size of a universal point set is quadratic and the best known lower bound is linear in n.
Some classes of planar graphs are known to admit universal point sets of near linear size, however, there are no truly linear bounds for interesting classes beyond outerplanar graphs.
In this paper, we show that there is a universal point set of size 2n-2 for the class of bipartite planar graphs with n vertices. The same point set is also universal for the class of n-vertex planar graphs of maximum degree 3. The point set used for the results is what we call an exploding double chain, and we prove that this point set allows planar straight-line embeddings of many more planar graphs, namely of all subgraphs of planar graphs admitting a one-sided Hamiltonian cycle.
The result for bipartite graphs also implies that every n-vertex plane graph has a 1-bend drawing all whose bends and vertices are contained in a specific point set of size 4n-6, this improves a bound of 6n-10 for the same problem by Löffler and Tóth.

Stefan Felsner, Hendrik Schrezenmaier, Felix Schröder, and Raphael Steiner. Linear Size Universal Point Sets for Classes of Planar Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{felsner_et_al:LIPIcs.SoCG.2023.31, author = {Felsner, Stefan and Schrezenmaier, Hendrik and Schr\"{o}der, Felix and Steiner, Raphael}, title = {{Linear Size Universal Point Sets for Classes of Planar Graphs}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.31}, URN = {urn:nbn:de:0030-drops-178814}, doi = {10.4230/LIPIcs.SoCG.2023.31}, annote = {Keywords: Graph drawing, Universal point set, One-sided Hamiltonian, 2-page book embedding, Separating decomposition, Quadrangulation, 2-tree, Subcubic planar graph} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We study reconfiguration of independent sets in interval graphs under the token sliding rule. We show that if two independent sets of size k are reconfigurable in an n-vertex interval graph, then there is a reconfiguration sequence of length 𝒪(k⋅ n²). We also provide a construction in which the shortest reconfiguration sequence is of length Ω(k²⋅ n).
As a counterpart to these results, we also establish that Independent Set Reconfiguration is PSPACE-hard on incomparability graphs, of which interval graphs are a special case.

Marcin Briański, Stefan Felsner, Jędrzej Hodor, and Piotr Micek. Reconfiguring Independent Sets on Interval Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brianski_et_al:LIPIcs.MFCS.2021.23, author = {Bria\'{n}ski, Marcin and Felsner, Stefan and Hodor, J\k{e}drzej and Micek, Piotr}, title = {{Reconfiguring Independent Sets on Interval Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.23}, URN = {urn:nbn:de:0030-drops-144633}, doi = {10.4230/LIPIcs.MFCS.2021.23}, annote = {Keywords: reconfiguration, independent sets, interval graphs} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

The Koebe-Andreev-Thurston Circle Packing Theorem states that every triangulated planar graph has a contact representation by circles. The theorem has been generalized in various ways. The most prominent generalization assures the existence of a primal-dual circle representation for every 3-connected planar graph. We present a simple and elegant elementary proof of this result.

Stefan Felsner and Günter Rote. On Primal-Dual Circle Representations. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{felsner_et_al:OASIcs.SOSA.2019.8, author = {Felsner, Stefan and Rote, G\"{u}nter}, title = {{On Primal-Dual Circle Representations}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {8:1--8:18}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.8}, URN = {urn:nbn:de:0030-drops-100349}, doi = {10.4230/OASIcs.SOSA.2019.8}, annote = {Keywords: Disk packing, planar graphs, contact representation} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

The flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane spanning trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1,2,...,n } and the flip graph of k-element subsets of {1,2,...,n }. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k.

Stefan Felsner, Linda Kleist, Torsten Mütze, and Leon Sering. Rainbow Cycles in Flip Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{felsner_et_al:LIPIcs.SoCG.2018.38, author = {Felsner, Stefan and Kleist, Linda and M\"{u}tze, Torsten and Sering, Leon}, title = {{Rainbow Cycles in Flip Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {38:1--38:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina 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/LIPIcs.SoCG.2018.38}, URN = {urn:nbn:de:0030-drops-87514}, doi = {10.4230/LIPIcs.SoCG.2018.38}, annote = {Keywords: flip graph, cycle, rainbow, Gray code, triangulation, spanning tree, matching, permutation, subset, combination} }

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

A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs depend on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex.

Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, and Manfred Scheucher. Strongly Monotone Drawings of Planar Graphs. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{felsner_et_al:LIPIcs.SoCG.2016.37, author = {Felsner, Stefan and Igamberdiev, Alexander and Kindermann, Philipp and Klemz, Boris and Mchedlidze, Tamara and Scheucher, Manfred}, title = {{Strongly Monotone Drawings of Planar Graphs}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {37:1--37:15}, 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.2016.37}, URN = {urn:nbn:de:0030-drops-59292}, doi = {10.4230/LIPIcs.SoCG.2016.37}, annote = {Keywords: graph drawing, planar graphs, strongly monotone, strictly convex, primal-dual circle packing} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(w^3) colors on graphs with maximum clique size w.
In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when w=2.
The left-of relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class C of posets obtained from convex sets between two lines with some other classes of posets: all 2-dimensional posets and all posets of height 2 are in C but there is a 3-dimensional poset of height 3 that does not belong to C.
We also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.

Stefan Felsner, Piotr Micek, and Torsten Ueckerdt. On-line Coloring between Two Lines. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 630-641, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{felsner_et_al:LIPIcs.SOCG.2015.630, author = {Felsner, Stefan and Micek, Piotr and Ueckerdt, Torsten}, title = {{On-line Coloring between Two Lines}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {630--641}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars 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/LIPIcs.SOCG.2015.630}, URN = {urn:nbn:de:0030-drops-50915}, doi = {10.4230/LIPIcs.SOCG.2015.630}, annote = {Keywords: intersection graphs, cocomparability graphs, on-line coloring} }

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