Document

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

The problem of deciding whether a biconnected planar digraph G = (V,E) can be augmented to become an st-planar graph by adding a set of oriented edges E' ⊆ V × V is known to be NP-complete. We show that the problem is fixed-parameter tractable when parameterized by the size of the set E'.

Liana Khazaliya, Philipp Kindermann, Giuseppe Liotta, Fabrizio Montecchiani, and Kirill Simonov. The st-Planar Edge Completion Problem Is Fixed-Parameter Tractable. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{khazaliya_et_al:LIPIcs.ISAAC.2023.46, author = {Khazaliya, Liana and Kindermann, Philipp and Liotta, Giuseppe and Montecchiani, Fabrizio and Simonov, Kirill}, title = {{The st-Planar Edge Completion Problem Is Fixed-Parameter Tractable}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {46:1--46:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.46}, URN = {urn:nbn:de:0030-drops-193483}, doi = {10.4230/LIPIcs.ISAAC.2023.46}, annote = {Keywords: st-planar graphs, parameterized complexity, upward planarity} }

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**Published in:** Dagstuhl Reports, Volume 13, Issue 4 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 23162 "New Frontiers of Parameterized Complexity in Graph Drawing”. The seminar was held in-person from April 16 to April 21, 2023. It brought together 32 researchers from the Graph Drawing and the Parameterized Complexity research communities to discuss and explore new research frontiers on the interface between the two fields. The report collects the abstracts of talks and open problems presented in the seminar, as well as brief progress reports from the working groups.

Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, Meirav Zehavi, and Liana Khazaliya. New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162). In Dagstuhl Reports, Volume 13, Issue 4, pp. 58-97, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{ganian_et_al:DagRep.13.4.58, author = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav and Khazaliya, Liana}, title = {{New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162)}}, pages = {58--97}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {13}, number = {4}, editor = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav and Khazaliya, Liana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.13.4.58}, URN = {urn:nbn:de:0030-drops-192393}, doi = {10.4230/DagRep.13.4.58}, annote = {Keywords: algorithm design, computational geometry, graph drawing, parameterized complexity} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an st-orientation orients each edge of the input graph such that the resulting digraph is acyclic, and it contains a single source s and a single sink t. Computing an st-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an st-orientation with at most k transitive edges is more challenging and it was recently proven to be NP-hard already when k = 0. We strengthen this result by showing that the problem remains NP-hard even for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth.

Carla Binucci, Giuseppe Liotta, Fabrizio Montecchiani, Giacomo Ortali, and Tommaso Piselli. On the Parameterized Complexity of Computing st-Orientations with Few Transitive Edges. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{binucci_et_al:LIPIcs.MFCS.2023.18, author = {Binucci, Carla and Liotta, Giuseppe and Montecchiani, Fabrizio and Ortali, Giacomo and Piselli, Tommaso}, title = {{On the Parameterized Complexity of Computing st-Orientations with Few Transitive Edges}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.18}, URN = {urn:nbn:de:0030-drops-185524}, doi = {10.4230/LIPIcs.MFCS.2023.18}, annote = {Keywords: st-orientations, parameterized complexity, graph drawing} }

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

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.

Sujoy Bhore, Robert Ganian, Liana Khazaliya, Fabrizio Montecchiani, and Martin Nöllenburg. Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2023.18, author = {Bhore, Sujoy and Ganian, Robert and Khazaliya, Liana and Montecchiani, Fabrizio and N\"{o}llenburg, Martin}, title = {{Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {18:1--18:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.18}, URN = {urn:nbn:de:0030-drops-178689}, doi = {10.4230/LIPIcs.SoCG.2023.18}, annote = {Keywords: orthogonal drawings, bend minimization, extension problems, parameterized complexity} }

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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-dev.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} }

Document

**Published in:** Dagstuhl Reports, Volume 11, Issue 6 (2021)

This report documents the program and the outcomes of Dagstuhl Seminar 21293 "Parameterized Complexity in Graph Drawing". The seminar was held mostly in-person from July 18 to July 23, 2021. It brought together 28 researchers from the Graph Drawing and the Parameterized Complexity research communities with the aim to discuss and explore open research questions on the interface between the two fields. The report collects the abstracts of talks and open problems presented in the seminar, as well as brief progress reports from the working groups.

Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, and Meirav Zehavi. Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 21293). In Dagstuhl Reports, Volume 11, Issue 6, pp. 82-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Article{ganian_et_al:DagRep.11.6.82, author = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav}, title = {{Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 21293)}}, pages = {82--123}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2021}, volume = {11}, number = {6}, editor = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.11.6.82}, URN = {urn:nbn:de:0030-drops-155817}, doi = {10.4230/DagRep.11.6.82}, annote = {Keywords: exact computation, graph algorithms, graph drawing, parameterized complexity} }

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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-dev.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:** 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-dev.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)

Let G be a simple topological graph and let Gamma be a polyline drawing of G. We say that Gamma partially preserves the topology of G if it has the same external boundary, the same rotation system, and the same set of crossings as G. Drawing Gamma fully preserves the topology of G if the planarization of G and the planarization of Gamma have the same planar embedding. We show that if the set of crossing-free edges of G forms a connected spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Omega(sqrt{n}). Concerning drawings that fully preserve the topology, we show that if G has skewness k, it admits one such drawing with curve complexity at most 2k; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

Emilio Di Giacomo, Peter Eades, Giuseppe Liotta, Henk Meijer, and Fabrizio Montecchiani. Polyline Drawings with Topological Constraints. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 39:1-39:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{digiacomo_et_al:LIPIcs.ISAAC.2018.39, author = {Di Giacomo, Emilio and Eades, Peter and Liotta, Giuseppe and Meijer, Henk and Montecchiani, Fabrizio}, title = {{Polyline Drawings with Topological Constraints}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {39:1--39:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.39}, URN = {urn:nbn:de:0030-drops-99871}, doi = {10.4230/LIPIcs.ISAAC.2018.39}, annote = {Keywords: Topological graphs, graph drawing, curve complexity, skewness-k graphs, k-planar graphs} }

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

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.

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-dev.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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**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-dev.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 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

Therese Biedl, Giuseppe Liotta, and Fabrizio Montecchiani. On Visibility Representations of Non-Planar Graphs. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{biedl_et_al:LIPIcs.SoCG.2016.19, author = {Biedl, Therese and Liotta, Giuseppe and Montecchiani, Fabrizio}, title = {{On Visibility Representations of Non-Planar Graphs}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {19:1--19:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.19}, URN = {urn:nbn:de:0030-drops-59116}, doi = {10.4230/LIPIcs.SoCG.2016.19}, annote = {Keywords: Visibility Representations, 1-Planarity, Recognition Algorithm, Forbidden Configuration} }

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