Computational Geometry-Theory and Applications最新文献

{"title":"Packing d-dimensional balls into a d + 1-dimensional container","authors":"Helmut Alt ,&nbsp;Sergio Cabello ,&nbsp;Otfried Cheong ,&nbsp;Ji-won Park ,&nbsp;Nadja Seiferth","doi":"10.1016/j.comgeo.2025.102219","DOIUrl":"10.1016/j.comgeo.2025.102219","url":null,"abstract":"<div><div>In this article, we consider the problems of finding in <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span> dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of <em>d</em>-dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for <em>n</em> such balls, a container of volume <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></mfrac></mrow></msup><mo>)</mo></math></span> is always sufficient and sometimes necessary. As a byproduct, this implies that for <span><math><mi>d</mi><mo>⩾</mo><mn>2</mn></math></span> there is no finite size <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional convex body into which all <em>d</em>-dimensional unit-radius balls can be packed simultaneously.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102219"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145120133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal bound for PCA with outliers using higher-degree Voronoi diagrams","authors":"Sajjad Hashemian ,&nbsp;Mohammad Saeed Arvenaghi ,&nbsp;Ebrahim Ardeshir-Larijani","doi":"10.1016/j.comgeo.2025.102231","DOIUrl":"10.1016/j.comgeo.2025.102231","url":null,"abstract":"<div><div>In this paper, we introduce new algorithms for Principal Component Analysis (PCA) with outliers. Utilizing techniques from computational geometry, specifically higher-degree Voronoi diagrams, we navigate to the optimal subspace for PCA even in the presence of outliers. This approach achieves an optimal solution with time complexity of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mtext>poly</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>. Additionally, we present a randomized algorithm with complexity <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>log</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>δ</mi><mo>)</mo><mo>/</mo><mi>C</mi><mo>(</mo><mi>d</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>α</mi><mo>)</mo><mo>)</mo><mtext>poly</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>. Our approach leverages properties of high-dimensional spaces and the separation condition of outliers to efficiently recover the optimal subspace. Our results demonstrate that higher-degree Voronoi diagrams, combined with probabilistic subspace selection techniques, provide an effective and scalable solution for PCA with outliers.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102231"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Guarding points on a terrain by watchtowers","authors":"Byeonguk Kang ,&nbsp;Junhyeok Choi ,&nbsp;Jeesun Han ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102210","DOIUrl":"10.1016/j.comgeo.2025.102210","url":null,"abstract":"<div><div>We study the problem of guarding points on an <em>x</em>-monotone polygonal chain, called a terrain, using <em>k</em> watchtowers. A watchtower is a vertical segment whose bottom endpoint lies on the terrain. A point on the terrain is visible from a watchtower if the line segment connecting the point and the top endpoint of the watchtower does not cross the terrain. Given a sequence of point sites lying on a terrain, we aim to partition the sequence into <em>k</em> contiguous subsequences and place <em>k</em> watchtowers on the terrain such that every point site in a subsequence is visible from the same watchtower and the maximum length of the watchtowers is minimized. We present efficient algorithms for two variants of the problem.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102210"},"PeriodicalIF":0.4,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"m-Watchmen's routes in minbar and generalized minbar polygons","authors":"Rahmat Ghasemi ,&nbsp;Alireza Bagheri ,&nbsp;Anna Brötzner ,&nbsp;Fatemeh Keshavarz-Kohjerdi ,&nbsp;Faezeh Farivar ,&nbsp;Bengt J. Nilsson ,&nbsp;Christiane Schmidt","doi":"10.1016/j.comgeo.2025.102217","DOIUrl":"10.1016/j.comgeo.2025.102217","url":null,"abstract":"<div><div>We study the problem of multiple anchored watchman routes, where we are given <em>m</em> starting points for watchmen, and aim to find routes for all watchmen such that all points in a polygon are visible from at least one route. We consider the problem in Minbar polygons,<span><span>²</span></span> which are staircase polygons for which the floor of the staircase solely consists of one horizontal and one vertical edge, and in generalized Minbar polygons, which relaxes the definition of Minbar polygons, allowing for non-rectilinear edges. For Minbar polygons, we exhibit polynomial time algorithms to compute optimal solutions for both the min-max and the min-sum criteria. The min-max algorithm takes <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mo>+</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time, using <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> storage, and the min-sum algorithm takes <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>m</mi><mo>+</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mo>)</mo></math></span> time, also using <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> storage.</div><div>For generalized Minbar polygons, we prove NP-hardness for the min-sum and min-max criteria, and present approximation algorithms for both criteria: an <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>-approximation taking <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>4</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time for the min-sum criterion, and a <span><math><mo>(</mo><mi>π</mi><mo>+</mo><mn>3</mn><mo>)</mo></math></span>-approximation taking <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time for the min-max criterion.</div><div>Minbar polygons and the non-rectilinear generalization of them may seem to be very restricted polygon classes but they form an adjacent pair where the multiple anchored watchman routes problem has a polynomial time solution in one class but is NP-hard in the slightly more generalized class. It is this property that motivates our study of these restricted polygon classes.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102217"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Big line or big convex polygon","authors":"David Conlon ,&nbsp;Jacob Fox ,&nbsp;Xiaoyu He ,&nbsp;Dhruv Mubayi ,&nbsp;Andrew Suk ,&nbsp;Jacques Verstraëte","doi":"10.1016/j.comgeo.2025.102218","DOIUrl":"10.1016/j.comgeo.2025.102218","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the minimum <em>N</em> such that every <em>N</em>-element point set in the plane contains either <em>ℓ</em> collinear members or <em>n</em> points in convex position. We prove that there is a constant <span><math><mi>C</mi><mo>&gt;</mo><mn>0</mn></math></span> such that, for each <span><math><mi>ℓ</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>,<span><span><span><math><mo>(</mo><mn>3</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>5</mn></mrow></msup><mo>&lt;</mo><mi>E</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>&lt;</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mi>C</mi><msqrt><mrow><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msqrt></mrow></msup><mo>.</mo></math></span></span></span> A similar extension of the well-known Erdős–Szekeres cups-caps theorem is also proved.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102218"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144913031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Piercing unit geodesic disks","authors":"Ahmad Biniaz ,&nbsp;Prosenjit Bose ,&nbsp;Thomas Shermer","doi":"10.1016/j.comgeo.2025.102209","DOIUrl":"10.1016/j.comgeo.2025.102209","url":null,"abstract":"<div><div>We prove that at most 3 points are always sufficient to pierce a set of <em>m</em> pairwise intersecting unit geodesic disks inside a simple polygon <em>P</em> with <em>n</em> vertices of which <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are reflex. We provide an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></math></span>-time algorithm to compute these at most 3 piercing points. Our bound is tight since it is known that in certain cases 3 points are necessary.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102209"},"PeriodicalIF":0.4,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algorithms for computing closest points for segments","authors":"Haitao Wang","doi":"10.1016/j.comgeo.2025.102196","DOIUrl":"10.1016/j.comgeo.2025.102196","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points and a set <em>S</em> of <em>n</em> segments in the plane, we consider the problem of computing for each segment of <em>S</em> its closest point in <em>P</em>. The previously best algorithm solves the problem in <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> has also been proved. In this paper, we present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> time algorithm, which matches the above lower bound. In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in <em>P</em>. Our new results improve the previous work.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102196"},"PeriodicalIF":0.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Growth rates of the number of empty triangles and simplices","authors":"Bhaswar B. Bhattacharya ,&nbsp;Sandip Das ,&nbsp;Sk Samim Islam ,&nbsp;Saumya Sen","doi":"10.1016/j.comgeo.2025.102197","DOIUrl":"10.1016/j.comgeo.2025.102197","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, in general position, denote by <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo></math></span> the number of empty triangles with vertices in <em>P</em>. In this paper we investigate by how much <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo></math></span> changes if a point <em>x</em> is removed from <em>P</em>. By constructing a graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> based on the arrangement of the empty triangles incident on <em>x</em>, we transform this geometric problem to the problem of counting triangles in the graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>. We study properties of the graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and, in particular, show that it is diamond-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa–Szemerédi problem. We also derive similar bounds for the growth rate of the number of empty simplices for point sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102197"},"PeriodicalIF":0.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"VHS: A package for homological simplification of voxelized plant root data for skeletonization","authors":"Erin W. Chambers ,&nbsp;Tao Ju ,&nbsp;David Letscher ,&nbsp;Hannah Schreiber ,&nbsp;Dan Zeng","doi":"10.1016/j.comgeo.2025.102198","DOIUrl":"10.1016/j.comgeo.2025.102198","url":null,"abstract":"<div><div>In this work, we present VHS (<strong>V</strong>oxelized <strong>H</strong>omological <strong>S</strong>implification), a C++ package whose purpose is to de-noise voxelized data and output a topologically accurate simplified shape. In contrast to previous work on voxelized homological simplification tools, our main goal is offering a better starting point for computing curve skeletons for shape analysis. This goal necessitates additional simplification beyond what other packages provide, although our approach extends and improves prior work on heuristic methods which compute approximate solutions for the homological simplification problem. Our tool is designed for and tested on voxelized plant roots, although it is potentially useful beyond this data set. While the homological simplification problem is NP-hard in general, our package is able to simplify almost all of the topological noise when used on data from plant root systems. Compared with existing simplification tools, our method strikes a better balance between topological simplicity and geometric accuracy, resulting in higher usability of the resulting skeletons. Our code is publicly available at <span><span>https://github.com/davidletscher/VHS/</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102198"},"PeriodicalIF":0.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the rectilinear crossing number of complete balanced multipartite graphs and balanced layered graphs","authors":"Ruy Fabila-Monroy ,&nbsp;Rosna Paul ,&nbsp;Jenifer Viafara-Chanchi ,&nbsp;Alexandra Weinberger","doi":"10.1016/j.comgeo.2025.102199","DOIUrl":"10.1016/j.comgeo.2025.102199","url":null,"abstract":"<div><div>A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments and its vertices are points in general position. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let <span><math><mi>n</mi><mo>≥</mo><mi>r</mi></math></span> be positive integers. The graph <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, is the complete balanced <em>r</em>-partite graph on <em>n</em> vertices, in which every set of the partition has at least <span><math><mo>⌊</mo><mi>n</mi><mo>/</mo><mi>r</mi><mo>⌋</mo></math></span> vertices. The balanced layered graph, <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, is an <em>r</em>-partite graph on <em>n</em> vertices, where <em>n</em> is multiple of <em>r</em>. Every partition of <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> contains <span><math><mi>n</mi><mo>/</mo><mi>r</mi></math></span> vertices; for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi><mo>−</mo><mn>1</mn></math></span>, all the vertices in the <em>i</em>-th partition are adjacent to all the vertices in the <span><math><mo>(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-th partition, and these are the only edges of <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>. In this paper, we give upper bounds on the rectilinear crossing numbers of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102199"},"PeriodicalIF":0.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143904056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}