Computational Geometry-Theory and Applications最新文献

{"title":"Online dominating set and coloring for geometric intersection graphs","authors":"Minati De ,&nbsp;Sambhav Khurana ,&nbsp;Satyam Singh","doi":"10.1016/j.comgeo.2026.102256","DOIUrl":"10.1016/j.comgeo.2026.102256","url":null,"abstract":"<div><div>We study the online minimum dominating set and minimum coloring problems in the context of geometric intersection graphs. We consider a graph parameter: the independent kissing number <em>ζ</em>, which is the number equal to “the size of the largest induced star in the graph −1”. For a graph with an independent kissing number of <em>ζ</em>, we show that the famous greedy algorithm achieves an optimal competitive ratio of <em>ζ</em> for the minimum dominating set and the minimum independent dominating set problems. However, for the minimum connected dominating set problem, we obtain a competitive ratio of at most 2<em>ζ</em>. To complement this, we prove that for the minimum connected dominating set problem, any deterministic online algorithm achieves a competitive ratio of at least <span><math><mn>2</mn><mo>(</mo><mi>ζ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, for the geometric intersection graph of translates of a convex object in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Next, for the minimum coloring problem, we obtain an algorithm with a competitive ratio of <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>log</mi><mo>⁡</mo><mi>m</mi><mo>)</mo></mrow></math></span> for geometric intersection graphs of <em>α</em>-fat objects in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> having widths in <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>m</mi><mo>]</mo></math></span>, where <span><math><msup><mrow><mi>ζ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is the independent kissing number of the geometric intersection graph of <em>α</em>-fat objects having widths in <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. Finally, we investigate the value of <em>ζ</em> for geometric intersection graphs of various families of geometric objects.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"134 ","pages":"Article 102256"},"PeriodicalIF":0.7,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146161961","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 right angles crossing graphs","authors":"Franz J. Brandenburg","doi":"10.1016/j.comgeo.2026.102255","DOIUrl":"10.1016/j.comgeo.2026.102255","url":null,"abstract":"<div><div>A graph is an <em>optimal right angle crossing graph</em> (also called an optimal RAC graph for short) if it has n vertices and 4n–10 edges and admits a straight-line drawing in the plane such that each edge is crossed at most once and edges cross only at a right angle. This implies that the drawing is <em>3T-</em> or <em>TTX-framed</em>, that is, the outer face is a triangle that is adjacent to three triangles or to two triangles and a crossing. An optimal <em>pseudo-RAC graph</em> is the topological version of an optimal RAC graph, where the restrictions to straight-line edges and right angle crossings are dropped.</div><div>We show that every 3T-framed optimal pseudo-RAC graph is an optimal RAC graph, that is, 3T-framed optimal pseudo-RAC embeddings can be stretched and orthogonalized. This is not true for TTX-framed embeddings. There are <em>n</em>-vertex 3T- and TTX-framed optimal RAC graphs for every <span><math><mi>n</mi><mo>≥</mo><mn>9</mn></math></span>, and eleven optimal RAC and fourteen optimal pseudo-RAC graphs with at most eight vertices. Optimal pseudo-RAC graphs can be recognized in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> time, where the recognition algorithm demonstrates that every optimal pseudo-RAC graph has at most three 1-planar embeddings, in which edges are crossed at most once.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"134 ","pages":"Article 102255"},"PeriodicalIF":0.7,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146161879","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":"Distinct distances between a line and strip","authors":"Sanjana Das ,&nbsp;Adam Sheffer","doi":"10.1016/j.comgeo.2026.102243","DOIUrl":"10.1016/j.comgeo.2026.102243","url":null,"abstract":"<div><div>We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances between points on two lines or curves. Unlike these existing problems, this new variant only makes sense if the points satisfy an additional spacing condition.</div><div>Our work can also be seen as an exploration of the proximity technique that was recently introduced by Solymosi and Zahl. This technique lies at the heart of our analysis.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102243"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977137","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":"Inscribed and circumscribed histogons of a convex polygon","authors":"Jaehoon Chung ,&nbsp;Sang Won Bae ,&nbsp;Chan-Su Shin ,&nbsp;Sang Duk Yoon ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102232","DOIUrl":"10.1016/j.comgeo.2025.102232","url":null,"abstract":"<div><div>We consider two optimization problems of approximating a convex polygon in the plane, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation <em>θ</em> is a copy of an axis-aligned histogon rotated by <em>θ</em> in a counterclockwise direction. Our goal is to compute a largest inscribed histogon and a smallest circumscribed histogon over all orientations in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>)</mo></math></span>. Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms for these versions of the inscribed histogon problem. For the circumscribed histogon problem, we present an efficient algorithm whose running time depends on the diameter and the number of vertices of the input polygon. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102232"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571941","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":"Rectangular drawing of cubic graphs on an annulus and a Möbius band","authors":"Atsuhiro Nakamoto ,&nbsp;Kyosuke Wakayama","doi":"10.1016/j.comgeo.2025.102242","DOIUrl":"10.1016/j.comgeo.2025.102242","url":null,"abstract":"<div><div>The rectangular drawing on the plane is a drawing of a plane graph which satisfies the following geometric conditions; each edge is represented as a horizontal or vertical line segment, and the boundary of each face has exactly four corners with angle <span><math><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. The necessary and sufficient condition for a subcubic plane graph to have a rectangular drawing was already given by Rahman et al. <span><span>[14]</span></span>. This result was extended to a drawing of graphs on an annulus by Hasheminezhad et al. <span><span>[5]</span></span>. However, in this paper, we point out an error in the result and get a necessary and sufficient condition for the graph to have a rectangular drawing on an annulus. By a simple analogy, we also obtain a similar result for the rectangular drawing on a Möbius band.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102242"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925040","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":"Maximum number of points in general position in a random subset of finite 3-dimensional spaces","authors":"József Balogh ,&nbsp;Haoran Luo","doi":"10.1016/j.comgeo.2026.102244","DOIUrl":"10.1016/j.comgeo.2026.102244","url":null,"abstract":"<div><div>Let <span><math><mi>α</mi><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>,</mo><mi>p</mi><mo>)</mo></math></span> be the maximum possible size of a point set in general position in the <em>p</em>-random subset of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>. In this note, we determine the order of magnitude of <span><math><mi>α</mi><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>,</mo><mi>p</mi><mo>)</mo></math></span> up to a polylogarithmic factor by proving a balanced supersaturation result for the sets of 4 points in the same plane.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102244"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977136","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":"Maximizing the maximum degree in ordered nearest neighbor graphs","authors":"Péter Ágoston ,&nbsp;Adrian Dumitrescu ,&nbsp;Arsenii Sagdeev ,&nbsp;Karamjeet Singh ,&nbsp;Ji Zeng","doi":"10.1016/j.comgeo.2025.102229","DOIUrl":"10.1016/j.comgeo.2025.102229","url":null,"abstract":"<div><div>For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the <em>ordered Nearest Neighbor Graph</em> is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least <span><math><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>/</mo><mo>(</mo><mn>4</mn><mi>d</mi><mo>)</mo></math></span>. Apart from the <span><math><mn>1</mn><mo>/</mo><mo>(</mo><mn>4</mn><mi>d</mi><mo>)</mo></math></span> factor, this bound is the best possible. As for the abstract setting, we show that for every <em>n</em>-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msqrt><mo>)</mo></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102229"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145120134","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":"Approximation of MWIS on geometric intersection graphs","authors":"C.R. Subramanian","doi":"10.1016/j.comgeo.2025.102228","DOIUrl":"10.1016/j.comgeo.2025.102228","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We present a generic formulation of an algorithmic paradigm for approximating maximum weighted independent sets (MWIS) in arbitrary vertex weighted graphs. A special case of this paradigm has been proposed earlier for geometric intersection graphs. Here, we propose and analyse a much more general formulation. As part of this formulation, we introduce a new graph parameter which plays a role in bounding the approximation factor of the algorithms. By applying this paradigm to intersection graph classes of specific types of geometric objects, we obtain efficient algorithms which approximate a MWIS within &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; multiplicative factors. It is also shown that the same approach can be generalised to obtain efficient approximation algorithms for computing an optimal weight &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-subgraphs where &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a suitable hereditary property.&lt;/div&gt;&lt;div&gt;Applying our paradigm, we establish, for every &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, that MWIS of the intersection graph of a given collection of weighted &lt;em&gt;k&lt;/em&gt;-dimensional &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; spheres (having a common radius) can be efficiently approximated within a multiplicative factor of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. The running time can be brought down to &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; at the cost of increasing the approximation guarantee to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, for some constant &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; depending only on &lt;em&gt;p&lt;/em&gt; and &lt;em&gt;k&lt;/em&gt;. It is also shown that the above MWIS-approximation results can be extended to MWIS-approximation over the more general intersection graphs of finite collections of connected, full-dimensional and centrally-symmetric bodies in &lt;em&gt;k&lt;/em&gt;-dimensional, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-spaces, for every &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;In a related development, we also establish the following graph theoretic result which will be of independent interest: For every &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102228"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099437","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 structure of extremal point-line arrangements","authors":"Gabriel Currier ,&nbsp;Jozsef Solymosi ,&nbsp;Hung-Hsun Hans Yu","doi":"10.1016/j.comgeo.2025.102227","DOIUrl":"10.1016/j.comgeo.2025.102227","url":null,"abstract":"<div><div>In this note, we show that extremal Szemerédi–Trotter configurations are rigid in the following sense: If <span><math><mi>P</mi><mo>,</mo><mi>L</mi></math></span> are sets of points and lines determining at least <span><math><mi>C</mi><mo>|</mo><mi>P</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>|</mo><mi>L</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> incidences, then there exists a collection <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of points of size at most <span><math><mi>k</mi><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth–Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester–Gallai configurations.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102227"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099438","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":"City guarding with cameras of bounded field of view","authors":"Ahmad Biniaz,&nbsp;Mohammad Hashemi","doi":"10.1016/j.comgeo.2025.102230","DOIUrl":"10.1016/j.comgeo.2025.102230","url":null,"abstract":"<div><div>We study two problems related to the city guarding and the art gallery problems.<ul><li><span>1.</span><span><div>Given a city with <em>k</em> rectangular buildings, we prove that <span><math><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras of <span><math><msup><mrow><mn>180</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> field of view are always sufficient to guard the free space (the ground, walls, roofs, and the sky). This answers a conjecture of Daescu and Malik (2020) <span><span>[7]</span></span>.</div></span></li><li><span>2.</span><span><div>Given <em>k</em> orthogonally convex polygons of total <em>m</em> vertices in the plane, we prove that <span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras of <span><math><msup><mrow><mn>180</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> field of view are always sufficient to guard the free space (avoiding all the polygons). This answers another conjecture of Daescu and Malik (2021) <span><span>[8]</span></span>.</div></span></li></ul> Both upper bounds are tight in the sense that there are input instances that require these many cameras. Our proofs are constructive and suggest simple polynomial-time algorithms for placing these many cameras.</div><div>We then generalize the above bounds to arbitrary convex-shape buildings. We can guard the free space of <em>k</em> buildings of total size <em>m</em> by <span><math><mi>m</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras. For <em>k</em> simple polygons with <em>c</em> convex vertices in the plane we can guard the free space by <span><math><mi>c</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras. Again, both these bounds are tight.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102230"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222355","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}