Discrete & Computational Geometry最新文献

{"title":"Subword Complexes and Kalai's Conjecture on Reconstruction of Spheres.","authors":"Cesar Ceballos, Joseph Doolittle","doi":"10.1007/s00454-025-00733-6","DOIUrl":"https://doi.org/10.1007/s00454-025-00733-6","url":null,"abstract":"<p><p>A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani (Aequationes Math 34(2-3):287-297, 1987, 10.1007/BF01830678), via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai (2009, https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/). In this paper, we show that Kalai's conjecture holds in the particular case of Knutson and Miller's spherical subword complexes. This family of simplicial spheres arises in the context of Coxeter groups, and is conjectured to be polytopal. In contrast, not all manifolds are reconstructible. We show two explicit examples, namely the torus and the projective plane.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 1","pages":"23-48"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12177012/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144477722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The <ns0:math><ns0:mi>χ</ns0:mi></ns0:math> -Binding Function of <i>d</i>-Directional Segment Graphs.","authors":"Lech Duraj, Ross J Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald","doi":"10.1007/s00454-025-00737-2","DOIUrl":"https://doi.org/10.1007/s00454-025-00737-2","url":null,"abstract":"<p><p>Given a positive integer <i>d</i>, the class <i>d</i>-DIR is defined as all those intersection graphs formed from a finite collection of line segments in <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> having at most <i>d</i> slopes. Since each slope induces an interval graph, it easily follows for every <i>G</i> in <i>d</i>-DIR with clique number at most <math><mi>ω</mi></math> that the chromatic number <math><mrow><mi>χ</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </math> of <i>G</i> is at most <math><mrow><mi>d</mi> <mi>ω</mi></mrow> </math> . We show for every even value of <math><mi>ω</mi></math> how to construct a graph in <i>d</i>-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the <math><mi>χ</mi></math> -binding function of <i>d</i>-DIR is <math><mrow><mi>ω</mi> <mo>↦</mo> <mi>d</mi> <mi>ω</mi></mrow> </math> for <math><mi>ω</mi></math> even and <math><mrow><mi>ω</mi> <mo>↦</mo> <mi>d</mi> <mo>(</mo> <mi>ω</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> <mo>+</mo> <mn>1</mn></mrow> </math> for <math><mi>ω</mi></math> odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case <math><mrow><mi>d</mi> <mo>=</mo> <mn>2</mn></mrow> </math> .</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 3","pages":"758-770"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12484362/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145214309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Compact Metric Spaces with Infinite Cop Number.","authors":"Agelos Georgakopoulos","doi":"10.1007/s00454-024-00696-0","DOIUrl":"https://doi.org/10.1007/s00454-024-00696-0","url":null,"abstract":"<p><p>Mohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and that their number can be upper-bounded in terms of the ranks of the homology groups when the space is a simplicial pseudo-manifold. We disprove these conjectures by constructing a metric on <math> <msup><mrow><mi>S</mi></mrow> <mn>3</mn></msup> </math> with infinite cop number. More problems are raised than settled.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 3","pages":"793-804"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12484412/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145214328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rigidity of Symmetric Frameworks on the Cylinder.","authors":"Anthony Nixon, Bernd Schulze, Joseph Wall","doi":"10.1007/s00454-025-00723-8","DOIUrl":"https://doi.org/10.1007/s00454-025-00723-8","url":null,"abstract":"<p><p>A bar-joint framework (<i>G</i>, <i>p</i>) is the combination of a finite simple graph <math><mrow><mi>G</mi> <mo>=</mo> <mo>(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo>)</mo></mrow> </math> and a placement <math><mrow><mi>p</mi> <mo>:</mo> <mi>V</mi> <mo>→</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> . The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in <math> <msup><mrow><mi>R</mi></mrow> <mn>3</mn></msup> </math> restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"73 3","pages":"629-673"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11914369/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143665211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Monochromatic Infinite Sets in Minkowski Planes.","authors":"Nóra Frankl, Panna Gehér, Arsenii Sagdeev, Géza Tóth","doi":"10.1007/s00454-024-00702-5","DOIUrl":"https://doi.org/10.1007/s00454-024-00702-5","url":null,"abstract":"<p><p>We prove that for any <math><msub><mi>ℓ</mi> <mi>p</mi></msub> </math> -norm in the plane with <math><mrow><mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> and for every infinite <math><mrow><mi>M</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </mrow> </math> , there exists a two-colouring of the plane such that no isometric copy of <math><mi>M</mi></math> is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite <math><mrow><mi>M</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </mrow> </math> such that for every two-colouring of the plane there exists a monochromatic isometric copy of <math><mi>M</mi></math> .</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 2","pages":"569-583"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12449424/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145114837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Complexity of Order Type Isomorphism","authors":"Greg Aloupis, John Iacono, Stefan Langerman, Özgür Özkan, Stefanie Wuhrer","doi":"10.1007/s00454-024-00687-1","DOIUrl":"https://doi.org/10.1007/s00454-024-00687-1","url":null,"abstract":"<p>The order type of a point set in <span>(mathbb {R}^d)</span> maps each <span>((d{+}1))</span>-tuple of points to its orientation (e.g., clockwise or counterclockwise in <span>(mathbb {R}^2)</span>). Two point sets <i>X</i> and <i>Y</i> have the same order type if there exists a bijection <i>f</i> from <i>X</i> to <i>Y</i> for which every <span>((d{+}1))</span>-tuple <span>((a_1,a_2,ldots ,a_{d+1}))</span> of <i>X</i> and the corresponding tuple <span>((f(a_1),f(a_2),ldots ,f(a_{d+1})))</span> in <i>Y</i> have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an <span>(O(n^d))</span> algorithm for this task, thereby improving upon the <span>(O(n^{lfloor {3d/2}rfloor }))</span> algorithm of Goodman and Pollack (SIAM J. Comput. 12(3):484–507, 1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.\u0000</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"141 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Volume Computation for Meissner Polyhedra and Applications","authors":"Beniamin Bogosel","doi":"10.1007/s00454-024-00688-0","DOIUrl":"https://doi.org/10.1007/s00454-024-00688-0","url":null,"abstract":"<p>The volume of a Meissner polyhedron is computed in terms of the lengths of its dual edges. This allows to reformulate the Meissner conjecture regarding constant width bodies with minimal volume as a series of explicit finite dimensional problems. A direct consequence is the minimality of the volume of Meissner tetrahedras among Meissner pyramids.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Erdős–Szekeres-Type Problems in the Real Projective Plane","authors":"Martin Balko, Manfred Scheucher, Pavel Valtr","doi":"10.1007/s00454-024-00691-5","DOIUrl":"https://doi.org/10.1007/s00454-024-00691-5","url":null,"abstract":"<p>We consider point sets in the real projective plane <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős–Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős–Szekeres theorem about point sets in convex position in <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span>, which was initiated by Harborth and Möller in 1994. The notion of convex position in <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> agrees with the definition of convex sets introduced by Steinitz in 1913. For <span>(k ge 3)</span>, an <i>(affine) </i><i>k</i>-<i>hole</i> in a finite set <span>(S subseteq {mathbb {R}}^2)</span> is a set of <i>k</i> points from <i>S</i> in convex position with no point of <i>S</i> in the interior of their convex hull. After introducing a new notion of <i>k</i>-holes for points sets from <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span>, called <i>projective </i><i>k</i>-<i>holes</i>, we find arbitrarily large finite sets of points from <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> with no projective 8-holes, providing an analogue of a classical planar construction by Horton from 1983. We also prove that they contain only quadratically many projective <i>k</i>-holes for <span>(k le 7)</span>. On the other hand, we show that the number of <i>k</i>-holes can be substantially larger in <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> than in <span>({mathbb {R}}^2)</span> by constructing, for every <span>(k in {3,dots ,6})</span>, sets of <i>n</i> points from <span>({mathbb {R}}^2 subset {{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> with <span>(Omega (n^{3-3/5k}))</span> projective <i>k</i>-holes and only <span>(O(n^2))</span> affine <i>k</i>-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> and about some algorithmic aspects. The study of extremal problems about point sets in <span>({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}})</span> opens a new area of research, which we support by posing several open problems.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Structure of Metrizable Graphs","authors":"Maria Chudnovsky, Daniel Cizma, Nati Linial","doi":"10.1007/s00454-024-00685-3","DOIUrl":"https://doi.org/10.1007/s00454-024-00685-3","url":null,"abstract":"<p>A <i>consistent path system</i> in a graph <i>G</i> is an intersection-closed collection of paths, with exactly one path between any two vertices in <i>G</i>. We call <i>G</i> <i>metrizable</i> if every consistent path system in it is the system of geodesic paths defined by assigning some positive lengths to its edges. We show that metrizable graphs are, in essence, subdivisions of a small family of basic graphs with additional compliant edges. In particular, we show that every metrizable graph with 11 vertices or more is outerplanar plus one vertex.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"24-25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications","authors":"Thomas Lew, Riccardo Bonalli, Lucas Janson, Marco Pavone","doi":"10.1007/s00454-024-00683-5","DOIUrl":"https://doi.org/10.1007/s00454-024-00683-5","url":null,"abstract":"<p>We study the problem of estimating the convex hull of the image <span>(f(X)subset {mathbb {R}}^n)</span> of a compact set <span>(Xsubset {mathbb {R}}^m)</span> with smooth boundary through a smooth function <span>(f:{mathbb {R}}^mrightarrow {mathbb {R}}^n)</span>. Assuming that <i>f</i> is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of <i>f</i>(<i>X</i>) and the convex hull of the images <span>(f(x_i))</span> of <i>M</i> sampled inputs <span>(x_i)</span> on the boundary of <i>X</i>. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}