Pub Date : 2024-07-27DOI: 10.4230/LIPIcs.SoCG.2024.27
K. Buchin, M. Buchin, Joachim Gudmundsson, Aleksandr Popov, Sampson Wong
Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fr'echet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fr'echet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [SODA 2023, arXiv:2211.02951] studied this problem for arbitrary query polygonal curves and $c$-packed graphs. In this paper, we instead require the graphs to be $lambda$-low-density $t$-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.
{"title":"Map-Matching Queries Under Fréchet Distance on Low-Density Spanners","authors":"K. Buchin, M. Buchin, Joachim Gudmundsson, Aleksandr Popov, Sampson Wong","doi":"10.4230/LIPIcs.SoCG.2024.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2024.27","url":null,"abstract":"Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fr'echet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fr'echet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [SODA 2023, arXiv:2211.02951] studied this problem for arbitrary query polygonal curves and $c$-packed graphs. In this paper, we instead require the graphs to be $lambda$-low-density $t$-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"8 10","pages":"27:1-27:15"},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141797284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-04DOI: 10.4230/LIPIcs.SoCG.2024.3
Eyal Ackerman, G'abor Dam'asdi, Balázs Keszegh, R. Pinchasi, Rebeka Raffay
A long-standing open conjecture of Branko Gr"unbaum from 1972 states that any simple arrangement of $n$ pairwise intersecting pseudocircles in the plane can have at most $2n-2$ digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Gr"unbaum's conjecture is true for arrangements of pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Gr"unbaum for any simple arrangement of pairwise intersecting circles in the plane.
{"title":"On the Number of Digons in Arrangements of Pairwise Intersecting Circles","authors":"Eyal Ackerman, G'abor Dam'asdi, Balázs Keszegh, R. Pinchasi, Rebeka Raffay","doi":"10.4230/LIPIcs.SoCG.2024.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2024.3","url":null,"abstract":"A long-standing open conjecture of Branko Gr\"unbaum from 1972 states that any simple arrangement of $n$ pairwise intersecting pseudocircles in the plane can have at most $2n-2$ digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Gr\"unbaum's conjecture is true for arrangements of pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Gr\"unbaum for any simple arrangement of pairwise intersecting circles in the plane.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"226 1","pages":"3:1-3:14"},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141387023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-29DOI: 10.48550/arXiv.2303.16303
Timothy M. Chan, Zhengcheng Huang
In SoCG 2022, Conroy and T'oth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(nlog n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(nlog^2 n)$-size 3-hop spanner for $n$ axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an $O(1)$-hop spanner for arbitrary string graphs with $O(nalpha_k(n))$ size for any constant $k$, where $alpha_k(n)$ denotes the $k$-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an $O(1)$-hop spanner for intersection graphs of $d$-dimensional fat objects with $O(nalpha_k(n))$ size for any constant $k$ and $d$. We also improve on some of Conroy and T'oth's specific previous results, in either the number of hops or the size: we describe an $O(nlog n)$-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an $O(nlog n)$-size 3-hop spanner for axis-aligned rectangles in the plane. Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.
{"title":"Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size","authors":"Timothy M. Chan, Zhengcheng Huang","doi":"10.48550/arXiv.2303.16303","DOIUrl":"https://doi.org/10.48550/arXiv.2303.16303","url":null,"abstract":"In SoCG 2022, Conroy and T'oth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(nlog n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(nlog^2 n)$-size 3-hop spanner for $n$ axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an $O(1)$-hop spanner for arbitrary string graphs with $O(nalpha_k(n))$ size for any constant $k$, where $alpha_k(n)$ denotes the $k$-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an $O(1)$-hop spanner for intersection graphs of $d$-dimensional fat objects with $O(nalpha_k(n))$ size for any constant $k$ and $d$. We also improve on some of Conroy and T'oth's specific previous results, in either the number of hops or the size: we describe an $O(nlog n)$-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an $O(nlog n)$-size 3-hop spanner for axis-aligned rectangles in the plane. Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115614501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-20DOI: 10.4230/LIPIcs.SoCG.2023.15
Ulrich Bauer, Fabian Lenzen, M. Lesnick
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris--Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter PH. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter PH that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.
{"title":"Efficient Two-Parameter Persistence Computation via Cohomology","authors":"Ulrich Bauer, Fabian Lenzen, M. Lesnick","doi":"10.4230/LIPIcs.SoCG.2023.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2023.15","url":null,"abstract":"Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris--Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter PH. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter PH that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125567055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-17DOI: 10.48550/arXiv.2303.09702
A. Lubiw, Anurag Murty Naredla
The geodesic edge center of a polygon is a point c inside the polygon that minimizes the maximum geodesic distance from c to any edge of the polygon, where geodesic distance is the shortest path distance inside the polygon. We give a linear-time algorithm to find a geodesic edge center of a simple polygon. This improves on the previous O(n log n) time algorithm by Lubiw and Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an algorithm to find the geodesic vertex center of a simple polygon due to Pollack, Sharir, and Rote [Discrete&Computational Geometry, 1989] and an improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh [Discrete&Computational Geometry, 2016]. The geodesic edge center can easily be found from the geodesic farthest-edge Voronoi diagram of the polygon. Finding that Voronoi diagram in linear time is an open question, although the geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear time. As a first step of our geodesic edge center algorithm, we give a linear-time algorithm to find the geodesic farthest-edge Voronoi diagram restricted to the polygon boundary.
{"title":"The geodesic edge center of a simple polygon","authors":"A. Lubiw, Anurag Murty Naredla","doi":"10.48550/arXiv.2303.09702","DOIUrl":"https://doi.org/10.48550/arXiv.2303.09702","url":null,"abstract":"The geodesic edge center of a polygon is a point c inside the polygon that minimizes the maximum geodesic distance from c to any edge of the polygon, where geodesic distance is the shortest path distance inside the polygon. We give a linear-time algorithm to find a geodesic edge center of a simple polygon. This improves on the previous O(n log n) time algorithm by Lubiw and Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an algorithm to find the geodesic vertex center of a simple polygon due to Pollack, Sharir, and Rote [Discrete&Computational Geometry, 1989] and an improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh [Discrete&Computational Geometry, 2016]. The geodesic edge center can easily be found from the geodesic farthest-edge Voronoi diagram of the polygon. Finding that Voronoi diagram in linear time is an open question, although the geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear time. As a first step of our geodesic edge center algorithm, we give a linear-time algorithm to find the geodesic farthest-edge Voronoi diagram restricted to the polygon boundary.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"175 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126974924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-16DOI: 10.48550/arXiv.2303.09586
S. Arya, D. Mount
Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body $K$ of diameter $Delta$ in $textbf{R}^d$ for fixed $d$. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error $varepsilon$. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that $Theta((Delta/varepsilon)^{(d-1)/2})$ vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies. A natural way to characterize a convex object's skinniness is in terms of its relationship to the Euclidean ball. Given a convex body $K$, define its emph{volume diameter} $Delta_d$ to be the diameter of a Euclidean ball of the same volume as $K$, and define its emph{surface diameter} $Delta_{d-1}$ analogously for surface area. It follows from generalizations of the isoperimetric inequality that $Delta geq Delta_{d-1} geq Delta_d$. Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to $O((Delta_{d-1}/varepsilon)^{(d-1)/2})$. In this paper, we strengthen this by proving the existence of an approximation with $O((Delta_d/varepsilon)^{(d-1)/2})$ facets.
{"title":"Optimal Volume-Sensitive Bounds for Polytope Approximation","authors":"S. Arya, D. Mount","doi":"10.48550/arXiv.2303.09586","DOIUrl":"https://doi.org/10.48550/arXiv.2303.09586","url":null,"abstract":"Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body $K$ of diameter $Delta$ in $textbf{R}^d$ for fixed $d$. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error $varepsilon$. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that $Theta((Delta/varepsilon)^{(d-1)/2})$ vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies. A natural way to characterize a convex object's skinniness is in terms of its relationship to the Euclidean ball. Given a convex body $K$, define its emph{volume diameter} $Delta_d$ to be the diameter of a Euclidean ball of the same volume as $K$, and define its emph{surface diameter} $Delta_{d-1}$ analogously for surface area. It follows from generalizations of the isoperimetric inequality that $Delta geq Delta_{d-1} geq Delta_d$. Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to $O((Delta_{d-1}/varepsilon)^{(d-1)/2})$. In this paper, we strengthen this by proving the existence of an approximation with $O((Delta_d/varepsilon)^{(d-1)/2})$ facets.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131326444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-15DOI: 10.48550/arXiv.2303.08726
M. Hoffmann, Meghana M. Reddy
A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains $2$-planar. A $2$-planar graph on $n$ vertices has at most $5n-10$ edges, and some (maximal) $2$-planar graphs -- referred to as optimal $2$-planar -- achieve this bound. However, in strong contrast to maximal planar graphs, a maximal $2$-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density of maximal $2$-planar graphs by proving that every maximal $2$-planar graph on $nge 5$ vertices has at least $2n$ edges. We also show that this bound is tight, up to an additive constant. The lower bound is based on an analysis of the degree distribution in specific classes of drawings of the graph. The upper bound construction is verified by carefully exploring the space of admissible drawings using computer support.
{"title":"The Number of Edges in Maximal 2-planar Graphs","authors":"M. Hoffmann, Meghana M. Reddy","doi":"10.48550/arXiv.2303.08726","DOIUrl":"https://doi.org/10.48550/arXiv.2303.08726","url":null,"abstract":"A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains $2$-planar. A $2$-planar graph on $n$ vertices has at most $5n-10$ edges, and some (maximal) $2$-planar graphs -- referred to as optimal $2$-planar -- achieve this bound. However, in strong contrast to maximal planar graphs, a maximal $2$-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density of maximal $2$-planar graphs by proving that every maximal $2$-planar graph on $nge 5$ vertices has at least $2n$ edges. We also show that this bound is tight, up to an additive constant. The lower bound is based on an analysis of the degree distribution in specific classes of drawings of the graph. The upper bound construction is verified by carefully exploring the space of admissible drawings using computer support.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121797877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-14DOI: 10.48550/arXiv.2303.08270
Nathaniel Clause, T. Dey, Facundo M'emoli, Bei Wang
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the M"{o}bius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module $M$ indexed by a bifiltration of $n$ simplices in $O(n^3)$ time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has $O(n^4)$ runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of $M$ from $O(n^4)$ to $O(n^3)$. In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.
{"title":"Meta-Diagrams for 2-Parameter Persistence","authors":"Nathaniel Clause, T. Dey, Facundo M'emoli, Bei Wang","doi":"10.48550/arXiv.2303.08270","DOIUrl":"https://doi.org/10.48550/arXiv.2303.08270","url":null,"abstract":"We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the M\"{o}bius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module $M$ indexed by a bifiltration of $n$ simplices in $O(n^3)$ time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has $O(n^4)$ runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of $M$ from $O(n^4)$ to $O(n^3)$. In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131024159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-14DOI: 10.48550/arXiv.2303.07982
C. Lunel, A. D. Mesmay
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.
{"title":"A Structural Approach to Tree Decompositions of Knots and Spatial Graphs","authors":"C. Lunel, A. D. Mesmay","doi":"10.48550/arXiv.2303.07982","DOIUrl":"https://doi.org/10.48550/arXiv.2303.07982","url":null,"abstract":"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.","PeriodicalId":413611,"journal":{"name":"International Symposium on Computational Geometry","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122958374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-13DOI: 10.48550/arXiv.2303.07401
O. Aichholzer, Man-Kwun Chiu, H. P. Hoang, M. Hoffmann, J. Kynčl, Yannic Maus, B. Vogtenhuber, A. Weinberger
For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph $K_n$ with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on $n$ vertices is bounded by $O(n^{16})$. The latter proof uses a Carath'eodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the sense that having the same ERS does not remain sufficient when removing or adding very few edges.
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