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

While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.

Pierre Dehornoy, Corentin Lunel, and Arnaud de Mesmay. Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dehornoy_et_al:LIPIcs.SoCG.2024.48, author = {Dehornoy, Pierre and Lunel, Corentin and de Mesmay, Arnaud}, title = {{Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {48:1--48:18}, 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.2024.48}, URN = {urn:nbn:de:0030-drops-199938}, doi = {10.4230/LIPIcs.SoCG.2024.48}, annote = {Keywords: Knot Theory, Genus, Slice Genus, Hopf Arborescent Links, Well-Quasi-Order} }

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

Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion.

Corentin Lunel and Arnaud de Mesmay. A Structural Approach to Tree Decompositions of Knots and Spatial Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lunel_et_al:LIPIcs.SoCG.2023.50, author = {Lunel, Corentin and de Mesmay, Arnaud}, title = {{A Structural Approach to Tree Decompositions of Knots and Spatial Graphs}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {50:1--50:16}, 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.2023.50}, URN = {urn:nbn:de:0030-drops-179002}, doi = {10.4230/LIPIcs.SoCG.2023.50}, annote = {Keywords: Knots, Spatial Graphs, Tree Decompositions, Tangle, Representativity} }

Document

**Published in:** Dagstuhl Reports, Volume 12, Issue 2 (2022)

This report documents the program and the outcomes of Dagstuhl Seminar 22062 "Computation and Reconfiguration in Low-Dimensional Topological Spaces". The seminar consisted of a small collection of introductory talks, an open problem session, and then the seminar participants worked in small groups on problems on reconfiguration within the context of objects as diverse as elimination trees, morphings, curves on surfaces, translation surfaces and Delaunay triangulations.

Maike Buchin, Anna Lubiw, Arnaud de Mesmay, Saul Schleimer, and Florestan Brunck. Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062). In Dagstuhl Reports, Volume 12, Issue 2, pp. 17-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{buchin_et_al:DagRep.12.2.17, author = {Buchin, Maike and Lubiw, Anna and de Mesmay, Arnaud and Schleimer, Saul and Brunck, Florestan}, title = {{Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062)}}, pages = {17--66}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {12}, number = {2}, editor = {Buchin, Maike and Lubiw, Anna and de Mesmay, Arnaud and Schleimer, Saul and Brunck, Florestan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.2.17}, URN = {urn:nbn:de:0030-drops-169305}, doi = {10.4230/DagRep.12.2.17}, annote = {Keywords: Geometric Topology, Computational Geometry, Graph Drawing, Reconfiguration, Dagstuhl Seminar} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most π on both sides. While the existence of a simple closed quasigeodesic on a convex polyhedron has been proved by Pogorelov in 1949, finding a polynomial-time algorithm to compute such a simple closed quasigeodesic has been repeatedly posed as an open problem. Our first contribution is to propose an extended definition of quasigeodesics in the intrinsic setting of (not necessarily convex) polyhedral spheres, and to prove the existence of a weakly simple closed quasigeodesic in such a setting. Our proof does not proceed via an approximation by smooth surfaces, but relies on an adapation of the disk flow of Hass and Scott to the context of polyhedral surfaces. Our second result is to leverage this existence theorem to provide a finite algorithm to compute a weakly simple closed quasigeodesic on a polyhedral sphere. On a convex polyhedron, our algorithm computes a simple closed quasigeodesic, solving an open problem of Demaine, Hersterberg and Ku.

Jean Chartier and Arnaud de Mesmay. Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chartier_et_al:LIPIcs.SoCG.2022.27, author = {Chartier, Jean and de Mesmay, Arnaud}, title = {{Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.27}, URN = {urn:nbn:de:0030-drops-160350}, doi = {10.4230/LIPIcs.SoCG.2022.27}, annote = {Keywords: Quasigeodesic, polyhedron, curve-shortening process, disk flow, weakly simple} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-Štefankovič with ideas coming from computational biology, specifically from the signed reversal distance algorithm of Hannenhalli-Pevzner. This result confirms a special case of a conjecture of Negami on the joint crossing number of two embeddable graphs. We also provide a correction for an argument of Negami bounding the joint crossing number of two non-orientable graph embeddings.

Niloufar Fuladi, Alfredo Hubard, and Arnaud de Mesmay. Short Topological Decompositions of Non-Orientable Surfaces. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fuladi_et_al:LIPIcs.SoCG.2022.41, author = {Fuladi, Niloufar and Hubard, Alfredo and de Mesmay, Arnaud}, title = {{Short Topological Decompositions of Non-Orientable Surfaces}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.41}, URN = {urn:nbn:de:0030-drops-160492}, doi = {10.4230/LIPIcs.SoCG.2022.41}, annote = {Keywords: Computational topology, embedded graph, non-orientable surface, joint crossing number, canonical system of loop, surface decomposition} }

Document

**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold.
As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve γ, and a collection of disjoint normal curves Δ, there is a polynomial-time algorithm to decide if γ lies in the normal subgroup generated by components of Δ in the fundamental group of the surface after attaching the curves to a basepoint.

Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, and Salman Parsa. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chambers_et_al:LIPIcs.SoCG.2021.23, author = {Chambers, Erin Wolf and Lazarus, Francis and de Mesmay, Arnaud and Parsa, Salman}, title = {{Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {23:1--23:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.23}, URN = {urn:nbn:de:0030-drops-138223}, doi = {10.4230/LIPIcs.SoCG.2021.23}, annote = {Keywords: 3-manifolds, surfaces, low-dimensional topology, contractibility, compressed curves} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem.
A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus g has a cut graph of length at most a given value. We prove a time lower bound for this problem of n^{Omega(g/log g)} conditionally to ETH. In other words, the first n^{O(g)}-time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year old question of these authors.
A multiway cut of an undirected graph G with t distinguished vertices, called terminals, is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n^{Omega(sqrt{gt + g^2}/log(gt))}, conditionally to ETH, for any choice of the genus g >=0 of the graph and the number of terminals t >=4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case).
Reductions to planar problems usually involve a grid-like structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value g of the genus.

Vincent Cohen-Addad, Éric Colin de Verdière, Dániel Marx, and Arnaud de Mesmay. Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2019.27, author = {Cohen-Addad, Vincent and Colin de Verdi\`{e}re, \'{E}ric and Marx, D\'{a}niel and de Mesmay, Arnaud}, title = {{Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.27}, URN = {urn:nbn:de:0030-drops-104311}, doi = {10.4230/LIPIcs.SoCG.2019.27}, annote = {Keywords: Cut graph, Multiway cut, Surface, Lower bound, Parameterized Complexity, Exponential Time Hypothesis} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number).

Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. The Unbearable Hardness of Unknotting. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{demesmay_et_al:LIPIcs.SoCG.2019.49, author = {de Mesmay, Arnaud and Rieck, Yo'av and Sedgwick, Eric and Tancer, Martin}, title = {{The Unbearable Hardness of Unknotting}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {49:1--49:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.49}, URN = {urn:nbn:de:0030-drops-104530}, doi = {10.4230/LIPIcs.SoCG.2019.49}, annote = {Keywords: Knot, Link, NP-hard, Reidemeister move, Unknot recognition, Unlinking number, intermediate invariants} }

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

We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds.
We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.

Benjamin A. Burton, Arnaud de Mesmay, and Uli Wagner. Finding Non-Orientable Surfaces in 3-Manifolds. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{burton_et_al:LIPIcs.SoCG.2016.24, author = {Burton, Benjamin A. and de Mesmay, Arnaud and Wagner, Uli}, title = {{Finding Non-Orientable Surfaces in 3-Manifolds}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {24:1--24:15}, 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.2016.24}, URN = {urn:nbn:de:0030-drops-59168}, doi = {10.4230/LIPIcs.SoCG.2016.24}, annote = {Keywords: 3-manifold, low-dimensional topology, embedding, non-orientability, normal surfaces} }

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

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property.
Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.

Alfredo Hubard, Vojtech Kaluža, Arnaud de Mesmay, and Martin Tancer. Shortest Path Embeddings of Graphs on Surfaces. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hubard_et_al:LIPIcs.SoCG.2016.43, author = {Hubard, Alfredo and Kalu\v{z}a, Vojtech and de Mesmay, Arnaud and Tancer, Martin}, title = {{Shortest Path Embeddings of Graphs on Surfaces}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {43:1--43:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.43}, URN = {urn:nbn:de:0030-drops-59356}, doi = {10.4230/LIPIcs.SoCG.2016.43}, annote = {Keywords: Graph embedding, surface, shortest path, crossing number, hyperbolic geometry} }

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