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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 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

In the classical Steiner tree problem, one is given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals T subseteq V. The objective is to find a minimum-cost edge set E' subseteq E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of rho = ln(4)+epsilon < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_1 subset ... subset T_k subseteq V, compute nested edge sets E_1 subseteq ... subseteq E_k subseteq E that span the corresponding terminal sets with minimum total cost.
The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2^k Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth. Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k <= 22), which works well for applications such as designing multi-level infrastructure or network visualization.

Reyan Ahmed, Patrizio Angelini, Faryad Darabi Sahneh, Alon Efrat, David Glickenstein, Martin Gronemann, Niklas Heinsohn, Stephen G. Kobourov, Richard Spence, Joseph Watkins, and Alexander Wolff. Multi-Level Steiner Trees. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ahmed_et_al:LIPIcs.SEA.2018.15, author = {Ahmed, Reyan and Angelini, Patrizio and Sahneh, Faryad Darabi and Efrat, Alon and Glickenstein, David and Gronemann, Martin and Heinsohn, Niklas and Kobourov, Stephen G. and Spence, Richard and Watkins, Joseph and Wolff, Alexander}, title = {{Multi-Level Steiner Trees}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {15:1--15:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.15}, URN = {urn:nbn:de:0030-drops-89506}, doi = {10.4230/LIPIcs.SEA.2018.15}, annote = {Keywords: Approximation algorithm, Steiner tree, multi-level graph representation} }

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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} }