Discrete and Computational Geometry最新文献

{"title":"Topological Art in Simple Galleries","authors":"Daniel Bertschinger, Nicolas El Maalouly, Tillmann Miltzow, Patrick Schnider, Simon Weber","doi":"10.1007/s00454-023-00540-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00540-x","url":null,"abstract":"Abstract Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P . We say two points $$a,bin P$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> can see each other if the line segment $${text {seg}} (a,b)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtext>seg</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is contained in P . We denote by V ( P ) the family of all minimum guard placements. The Hausdorff distance makes V ( P ) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V ( P ) is homotopy equivalent to S . Furthermore, for various concrete topological spaces T , we describe instances I of the art gallery problem such that V ( I ) is homeomorphic to T .","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"396 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135139355","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}

{"title":"A Tight Bound for the Number of Edges of Matchstick Graphs","authors":"Jérémy Lavollée, Konrad Swanepoel","doi":"10.1007/s00454-023-00530-z","DOIUrl":"https://doi.org/10.1007/s00454-023-00530-z","url":null,"abstract":"Abstract A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices is $$lfloor 3n-sqrt{12n-3}rfloor $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>12</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> . In this paper we prove this conjecture for all $$nge 1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The main geometric ingredient of the proof is an isoperimetric inequality related to L’Huilier’s inequality.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019995","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}

{"title":"Computing a Link Diagram From Its Exterior","authors":"Nathan M. Dunfield, Malik Obeidin, Cameron Gates Rudd","doi":"10.1007/s00454-023-00533-w","DOIUrl":"https://doi.org/10.1007/s00454-023-00533-w","url":null,"abstract":"Abstract A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136383403","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}

{"title":"The Crossing Tverberg Theorem","authors":"Radoslav Fulek, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, Uli Wagner","doi":"10.1007/s00454-023-00532-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00532-x","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135656160","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}

{"title":"Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres","authors":"Jean Chartier, Arnaud de Mesmay","doi":"10.1007/s00454-023-00511-2","DOIUrl":"https://doi.org/10.1007/s00454-023-00511-2","url":null,"abstract":"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 $$pi $$ 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 adaptation 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.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135400015","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}

{"title":"Fertilitopes","authors":"Colin Defant","doi":"10.1007/s00454-023-00488-y","DOIUrl":"https://doi.org/10.1007/s00454-023-00488-y","url":null,"abstract":"Abstract We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s . Associated to each permutation $$pi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>π</mml:mi> </mml:math> is a particular set $$mathcal V(pi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of integer compositions that appears in a formula for the fertility of $$pi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>π</mml:mi> </mml:math> , which is defined to be $$|s^{-1}(pi )|$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>s</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that $$mathcal V(pi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of $$pi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>π</mml:mi> </mml:math> to be the convex hull of $$mathcal V(pi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of $$pi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>π</mml:mi> </mml:math> directly from $$pi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>π</mml:mi> </mml:math> using Bousquet-Mélou’s notion of the canonical tree of $$pi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>π</mml:mi> </mml:math> . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees . We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that $$sum _{sigma in s^{-1}(pi )}x^{textrm{des}(sigma )+1}$$ <mml:math xmlns:mml=\"http://www.w3.org","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135363724","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}

{"title":"Finite 3-Orbit Polyhedra in Ordinary Space I","authors":"Gabe Cunningham, Daniel Pellicer","doi":"10.1007/s00454-023-00502-3","DOIUrl":"https://doi.org/10.1007/s00454-023-00502-3","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135643565","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}

{"title":"Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions","authors":"Gill Barequet, Evanthia Papadopoulou, Martin Suderland","doi":"10.1007/s00454-023-00492-2","DOIUrl":"https://doi.org/10.1007/s00454-023-00492-2","url":null,"abstract":"Abstract We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d -dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions $$mathbb {S}^{d-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . We show that the combinatorial complexity of the Gaussian map for the order- k Voronoi diagram of n line segments and lines is $$O(min {k,n-k}n^{d-1})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>min</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , which is tight for $$n-k=O(1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d -dimensional cells of the farthest Voronoi diagram are unbounded, its $$(d-1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -skeleton is connected, and it does not have tunnels. A d -cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of $$n ge 2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> lines in general position has exactly $$n(n-1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in $$O(n^{d-1} alpha (n))$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> time, for $$dge 4$$ <mml:math xmlns:mml=","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136248325","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}

{"title":"Maximum Cut on Interval Graphs of Interval Count Four is NP-Complete","authors":"Celina M. H. de Figueiredo, Alexsander A. de Melo, Fabiano S. Oliveira, Ana Silva","doi":"10.1007/s00454-023-00508-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00508-x","url":null,"abstract":"The computational complexity of the MaxCut problem restricted to interval graphs has been open since the 80’s, being one of the problems proposed by Johnson in his Ongoing Guide to NP-completeness, and has been settled as NP-complete only recently by Adhikary, Bose, Mukherjee, and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictive class of unit/proper interval graphs (or graphs with interval count 1) have been presented along the years, and the classification of the problem is still unknown. In this paper, we present the first NP-completeness proof for MaxCut when restricted to interval graphs with bounded interval count, namely graphs with interval count 4.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135160085","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}

{"title":"Nets in $$mathbb {P}^2$$ and Alexander Duality","authors":"Nancy Abdallah, Hal Schenck","doi":"10.1007/s00454-023-00504-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00504-1","url":null,"abstract":"A net in $$mathbb {P}^2$$ is a configuration of lines $$mathcal {A}$$ and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank $$le r$$ . In the context of line arrangements in $$mathbb {P}^2$$ , applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135326561","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}