Computing in Geometry and Topology
type: channel
description: Computing in Geometry and Topology aims to support the broader computational geometry and topology community by being a peer-reviewed scientific journal that provides diamond open access. The journal welcomes both applied and theoretical research in computational geometry and topology, as well as surveys. Computing in Geometry and Topology is sponsored by the Society for Computational Geometry.
publisher: Sponsored by the Society for Computational Geometry
language: en
publication name: Computing in Geometry and Topology
issn: 2750-7823
copyright: Authors who publish with this journal agree to the following terms: Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
items: Unit Edge-Length Rectilinear Drawings with Crossings and Rectangular Faces
items: Maximum Polygon Packing: The CG Challenge 2024
items: Shadoks Approach to Knapsack Polygonal Packing
items: Computing Maximum Polygonal Packings in Convex Polygons using Best-Fit, Genetic Algorithms and Integer Linear Programs
items: Guest Editors' Foreword
items: Bounds for \(k\)-centers of point sets under \(L_{\infty}\)-bottleneck distance
items: A near-linear time exact algorithm for the \(L_1\)-geodesic Fréchet distance between two curves on the boundary of a simple polygon
items: On the Crossing Number of Symmetric Configurations
items: Arrangements of Pseudocircles: On Digons and Triangles
items: Beyond Outerplanarity
items: Constructing Concise Convex Covers via Clique Covers
items: Sparsifying Disk Intersection Graphs for Reliable Connectivity
items: Finding maximum matchings in RDV graphs efficiently
items: The Tripartite-Circle Crossing Number of Graphs With Two Small Partition Classes
items: Outside-Obstacle Representations with All Vertices on the Outer Face
items: Hosting of the CG:SHOP Challenges
items: How Packed Is It, Really?
items: On Erdos-Szekeres Maker-Breaker games
items: An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs
items: The Mutual Visibility Problem for Fat Robots with Lights
items: Quadrangulating a Polygon with a Fixed Spirality using Steiner Points
items: Capturing the Shape of a Point Set With a Line Segment
items: Approximation Algorithms for Anchored Multiwatchman Routes
items: Approximating the Directed Hausdorff Distance
items: Convex Covering Using Collections of Convex Polygons and Set Cover
items: A Linear Time Algorithm for the Flip Distance between Plane Spanning Paths in Convex Point Sets
items: Pathwidth of 2-Layer k-Planar Graphs

Unit Edge-Length Rectilinear Drawings with Crossings and Rectangular Faces
type: item
description: Unit edge-length drawings, rectilinear drawings (where each edge is either a horizontal or a vertical segment), and rectangular face drawings are among the most studied subjects in Graph Drawing. However, most of the literature on these topics refers to planar graphs and planar drawings. In this paper we study drawings with all the above nice properties but that allow edge crossings; we call them Unit Edge-length Rectilinear drawings with Rectangular Faces (UER-RF drawings). We consider crossings as dummy vertices and apply the unit edge-length convention to the edge segments connecting any two (real or dummy) vertices. Note that UER-RF drawings are grid drawings (vertices are placed at distinct integer coordinates), which is another classical requirement of graph visualizations. We present several efficient and easily implementable algorithms for recognizing graphs that admit UER-RF drawings and for constructing such drawings if they exist. We consider restrictions on the degree of the vertices or on the size of the faces. For each type of restriction, we consider both the general unconstrained setting and a setting in which either the external boundary of the drawing is fixed or the rotation system of the graph is fixed as part of the input.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: EuroCG2025
  topic: category
creator: Patrizio Angelini
creator: Carla Binucci
creator: Giuseppe Di Battista
creator: Emilio Di Giacomo
creator: Walter Didimo
creator: Fabrizio Grosso
creator: Giacomo Ortali
creator: Ioannis G. Tollis
rights: Copyright (c) 2026 Patrizio Angelini, Carla Binucci, Giuseppe Di Battista, Emilio Di Giacomo, Walter Didimo, Fabrizio Grosso, Giacomo Ortali, Ioannis G. Tollis https://creativecommons.org/licenses/by/4.0
date: 2026-08-21
publication date: 2026-08-21
starting page: 4:1
ending page: 4:21
doi: 10.57717/cgt.v5i3.97

Maximum Polygon Packing: The CG Challenge 2024
type: item
description: We give an overview of the 2024 Computational Geometry Challenge targeting the problem Maximum Polygon Packing: Given a convex region \(P\) in the plane, and a collection of simple polygons \(Q_1, \ldots, Q_n\), each \(Q_i\) with a respective value \(c_i\), find a subset \(S \subseteq \{1, \ldots,n\}\) and a feasible packing within \(P\) of the polygons \(Q_i\) (without rotation) for \(i \in S\), maximizing \(\sum_{i \in S} c_i\). Geometric packing problems, such as this, present significant computational challenges and are of substantial practical importance.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: Surveys
  topic: category
subject: CG:SHOP2024
  topic: category
creator: Sándor P. Fekete
creator: Phillip Keldenich
creator: Dominik Krupke
creator: Stefan Schirra
rights: Copyright (c) 2026 Sándor P. Fekete, Phillip Keldenich, Dominik Krupke, Stefan Schirra https://creativecommons.org/licenses/by/4.0
date: 2026-08-20
publication date: 2026-08-20
starting page: 2:1
ending page: 2:16
doi: 10.57717/cgt.v5i5.125

Shadoks Approach to Knapsack Polygonal Packing
type: item
description: The 2024 edition of the CG:SHOP Challenge focused on the knapsack polygonal packing problem. Each instance consists of a convex polygon known as the container and a multiset of items, where each item is a simple polygon with an associated integer value. A feasible packing solution places a selection of the items inside the container without overlapping and using only translations. The goal is to achieve a packing that maximizes the total value of the items in the solution. Our approach to win first place is divided into two main steps. First, we generate promising initial solutions using two strategies: one based on integer linear programming and the other one employing a combination of geometric greedy heuristics. In the second step, we enhance these solutions through local search techniques, which involve repositioning items and exploring potential replacements to improve the total value of the packing.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2024
  topic: category
creator: Guilherme D. da Fonseca
creator: Yan Gerard
rights: Copyright (c) 2026 Guilherme D. da Fonseca, Yan Gerard https://creativecommons.org/licenses/by/4.0
date: 2026-08-20
publication date: 2026-08-20
starting page: 3:1
ending page: 3:14
doi: 10.57717/cgt.v5i5.80

Computing Maximum Polygonal Packings in Convex Polygons using Best-Fit, Genetic Algorithms and Integer Linear Programs
type: item
description: Given a convex region \(C\) and a set of simple polygons with associated profits, the Maximum Polygon Packing Problem seeks a non-overlapping packing of a subset of the polygons (without rotations) into \(C\), such that the total profit of the packed polygons is maximized. To handle instances of various sizes and properties, we present a collection of algorithms for this problem. For large instances, we utilize a greedy best-fit placement strategy. For instances consisting exclusively of rectilinear polygons, we use a specialized greedy best-fit algorithm which handles rectilinear shapes efficiently. For medium-sized instances, we provide a genetic algorithm. Finally, for the smallest instances, we employ an integer linear programming model to obtain near-optimal solutions.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2024
  topic: category
creator: Alkan Atak
creator: Kevin Buchin
creator: Mart Hagedoorn
creator: Jona Heinrichs
creator: Karsten Hogreve
creator: Guangping Li
creator: Patrick Pawelczyk
rights: Copyright (c) 2026 Alkan Atak, Kevin Buchin, Mart Hagedoorn, Jona Heinrichs, Karsten Hogreve, Guangping Li, Patrick Pawelczyk https://creativecommons.org/licenses/by/4.0
date: 2026-08-20
publication date: 2026-08-20
starting page: 4:1
ending page: 4:14
doi: 10.57717/cgt.v5i5.87

Guest Editors' Foreword
type: item
description: This is the guest editors’ foreword for the special issue on the Sixth CG Challenge, CG:SHOP 2024.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2024
  topic: category
creator: Sándor P. Fekete
creator: Stefan Schirra
rights: Copyright (c) 2026 Sándor P. Fekete, Stefan Schirra https://creativecommons.org/licenses/by/4.0
date: 2026-08-20
publication date: 2026-08-20
starting page: 1:1
ending page: 1:1
doi: 10.57717/cgt.v5i5.124

Bounds for \(k\)-centers of point sets under \(L_{\infty}\)-bottleneck distance
type: item
description: We consider the \(k\)-center problem on the space of fixed-size point sets in the plane under the \(L_{\infty}\)-bottleneck distance. While this problem is motivated by persistence diagrams in topological data analysis, we illustrate it as a Restaurant Supply Problem: given \(n\) restaurant chains of \(m\) stores each, we want to place supermarket chains, also of \(m\) stores each, such that each restaurant chain can select one supermarket chain to supply all its stores, ensuring that each store is matched to a nearby supermarket. How many supermarket chains are required to supply all restaurants? We address this questions under the constraint that any two restaurant chains are so close under the $L_{\infty}$-distance that they can be supplied by a single supermarket chain. We provide both upper and lower bounds for this problem and investigate its computational complexity.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: EuroCG2025
  topic: category
creator: Mats Bierwirth
creator: Julia Hütte
creator: Patrick Schnider
creator: Bettina Speckmann
rights: Copyright (c) 2026 Mats Bierwirth, Julia Hütte, Patrick Schnider, Bettina Speckmann https://creativecommons.org/licenses/by/4.0
date: 2026-08-16
publication date: 2026-08-16
starting page: 3:1
ending page: 3:14
doi: 10.57717/cgt.v5i3.91

A near-linear time exact algorithm for the \(L_1\)-geodesic Fréchet distance between two curves on the boundary of a simple polygon
type: item
description: Let \(P\) be a polygon with \(k\)vertices. Let \(R\) and \(B\) be two simple, interior disjoint curves on the boundary of \(P\), with \(n\) and \(m\) vertices. We show how to compute the Fréchet distance between \(R\) and \(B\) using the geodesic \(L_1\)-distance in \(P\) in \(O(k \log nm + (n+m) (\log^2 nm \log k + \log^4 nm))\) time.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Thijs van der Horst
creator: Marc van Kreveld
creator: Tim Ophelders
creator: Bettina Speckmann
rights: Copyright (c) 2022 Thijs van der Horst, Marc van Kreveld, Tim Ophelders, Bettina Speckmann https://creativecommons.org/licenses/by/4.0
date: 2026-08-12
publication date: 2026-08-12
starting page: 5:1
ending page: 5:20
doi: 10.57717/cgt.v5i1.114

On the Crossing Number of Symmetric Configurations
type: item
description: For any finite set of points P in general position in the plane, we consider the drawing of the complete graph with vertex set P , whose edges are the straight-line segments joining pairs of points. The crossing number of P is the number of pairs of segments that intersect in their interior. The minimum crossing number over all such sets P of n points is known as the rectilinear or geometric crossing number of the complete graph Kn . The problem of determining this crossing number was posed by Erdős and Guy in the early 1970s and, although it has been extensively studied, it remains open to this date with exact values known only for n ≤ 27 and n = 30. In 2010, it was conjectured that whenever the number of points is a multiple of 3, there are crossing optimal point sets with 3-fold (triangular) rotational symmetry. Motivated by this conjecture, this paper provides lower and upper bounds for sym-crm (Kn ), the rectilinear crossing number of Kn when the minimum is restricted to m-fold symmetric configurations of points in the plane. Our bounds are tight for even symmetry.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: CCCG2024
  topic: category
creator: Bernardo M. Ábrego
creator: Silvia Fernández-Merchant
rights: Copyright (c) 2026 Bernardo M. Ábrego, Silvia Fernández-Merchant https://creativecommons.org/licenses/by/4.0
date: 2026-06-02
publication date: 2026-06-02
starting page: 5:1
ending page: 5:28
doi: 10.57717/cgt.v5i2.75

Arrangements of Pseudocircles: On Digons and Triangles
type: item
description: In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Grünbaum (1972). First, we discuss the maximum number of digons or touching points. Grünbaum conjectured that there are at most 2n - 2 digon cells or equivalently at most 2n - 2 touchings. Agarwal et al. (2004) verified the conjecture for cylindrical arrangements. We show that the conjecture holds for any arrangement which contains three pairwise touching pseudocircles. The proof makes use of the result for cylindrical arrangements. Moreover, we construct non-cylindrical arrangements which attain the maximum of 2n - 2 touchings and have no triple of pairwise touching pseudocircles. Second, we discuss the minimum number of triangular cells (triangles) in arrangements without digons and touchings. Grünbaum conjectured that such arrangements have 2n - 4 triangles. Snoeyink and Hershberger (1991) established a lower bound of ⌈(4/3)n⌉. Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only ⌈(16/11)n⌉ triangles. We provide a construction which shows that ⌈(4/3)n⌉ is the correct value.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Stefan Felsner
creator: Sandro Roch
creator: Manfred Scheucher
rights: Copyright (c) 2024 Stefan Felsner, Sandro Roch, Manfred Scheucher https://creativecommons.org/licenses/by/4.0
date: 2024-12-19
publication date: 2024-12-19
starting page: 11:1
ending page: 11:18
doi: 10.57717/cgt.v3i1.67

Beyond Outerplanarity
type: item
description: We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with convex drawings: outer $k$-planar graphs, where each edge is crossed by at most k other edges; and outer k-quasi-planar graphs, where no k edges can mutually cross. We show that the outer k-planar graphs are ⌊3.5 √k⌋-degenerate, and consequently that every √outer k-planar graph can be colored with ⌊3.5 √k⌋ + 1 colors. We further show that every outer k-planar graph has a balanced vertex separator of size at most 2k + 3. For each fixed k, these small balanced separators allow us to test outer k-planarity in quasi-polynomial time, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis fails. We also show that the class of outer 3-quasi-planar graphs and the class of planar graphs are incomparable. Finally, we restrict outer k-planar and outer k-quasi-planar drawings to full drawings (where no crossing appears on the boundary of the outer face) and to closed drawings (where the vertex sequence on the boundary of the outer face is a Hamiltonian cycle in the graph). For each k, we express closed outer k-planarity and closed outer k-quasi-planarity in extended monadic second-order logic. Since every outer k-planar graph has treewidth O(k), Courcelle’s theorem implies that closed outer k-planarity is linear-time testable. We leverage this result to further show that full outer k-planarity can also be tested in linear time.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Computational Topology
  topic: category
creator: Steven Chaplick
creator: Myroslav Kryven
creator: Giuseppe Liotta
creator: Andre Löffler
creator: Alexander Wolff
rights: Copyright (c) 2026 Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, Alexander Wolff https://creativecommons.org/licenses/by/4.0
date: 2026-05-26
publication date: 2026-05-26
starting page: 4:1
ending page: 4:25
doi: 10.57717/cgt.v5i1.61

Constructing Concise Convex Covers via Clique Covers
type: item
description: This work is the full version describing the winning implementation of the CG:SHOP 2023 Challenge. The topic of the Challenge was the convex cover problem: given a polygon P (with holes), find a minimum-cardinality set of convex polygons whose union equals P. We use a three-step approach: Create a suitable triangulation of P. Compute a visibility graph of the triangles. Solve a vertex clique cover problem on the visibility graph, from which we then derive the convex cover. This way we capture the geometric difficulty in the first step and the combinatorial difficulty in the third step.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2023
  topic: category
creator: Mikkel Abrahamsen
creator: William Bille Meyling
creator: André Nusser
rights: Copyright (c) 2026 Mikkel Abrahamsen, William Bille Meyling, André Nusser https://creativecommons.org/licenses/by/4.0
date: 2026-02-19
publication date: 2026-02-19
starting page: 4:1
ending page: 4:15
doi: 10.57717/cgt.v5i4.76

Sparsifying Disk Intersection Graphs for Reliable Connectivity
type: item
description: The intersection graph induced by a set of n disks can be dense. It is thus natural to try and sparsify it, while preserving connectivity. Unfortunately, sparse graphs can always be made disconnected by removing a small number of vertices. In this work, we present a randomized sparsification algorithm that maintains connectivity between two regions in the computed graph, if the original graph remains "well-connected" even after removing an arbitrary `"attack'" set from both graphs. Thus, the new sparse graph has similar reliability to the original disk graph, and can withstand catastrophic failure of nodes while still providing a connectivity guarantee for the remaining graph. The new graph has near linear complexity, and can be constructed in near-linear time. The algorithm extends to any collection of shapes in the plane with near linear union complexity.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Eliot Robson
creator: Sariel Har-Peled
rights: Copyright (c) 2024 Eliot Robson, Sariel Har-Peled https://creativecommons.org/licenses/by/4.0
date: 2024-12-19
publication date: 2024-12-19
starting page: 10:1
ending page: 10:13
doi: 10.57717/cgt.v3i1.66

Finding maximum matchings in RDV graphs efficiently
type: item
description: In this paper, we study the maximum matching problem in RDV graphs, i.e., vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a horizontal segment intersects one of a dynamically changing set of vertical segments, which in turn reduces to a range minimum query. Using a suitable data structure, we can therefore find a maximum matching in O(n log n) time (presuming a linear-sized representation of the graph is given), i.e., without even looking at all edges.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: CCCG2024
  topic: category
creator: Therese Biedl
creator: Prashant Gokhale
rights: Copyright (c) 2026 Therese Biedl, Prashant Gokhale https://creativecommons.org/licenses/by/4.0
date: 2026-02-23
publication date: 2026-02-23
starting page: 4:1
ending page: 4:14
doi: 10.57717/cgt.v5i2.72

The Tripartite-Circle Crossing Number of Graphs With Two Small Partition Classes
type: item
description: A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of a tripartite graph is the minimum number of edge crossings among all its tripartite-circle drawings. We determine the exact value of the tripartite-circle crossing number of Ka,b,n, where a, b ≤ 2.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Computational Topology
  topic: category
creator: Charles Camacho
creator: Silvia Fernández-Merchant
creator: Marija Jelić Milutinović
creator: Rachel Kirsch
creator: Linda Kleist
creator: Elizabeth Matson
creator: Jennifer White
rights: Copyright (c) 2024 Charles Camacho, Silvia Fernández-Merchant, Marija Jelić Milutinović, Rachel Kirsch, Linda Kleist, Elizabeth Bailey Matson, Jennifer White https://creativecommons.org/licenses/by/4.0
date: 2024-11-07
publication date: 2024-11-07
starting page: 9:1
ending page: 9:21
doi: 10.57717/cgt.v3i1.63

Outside-Obstacle Representations with All Vertices on the Outer Face
type: item
description: An obstacle representation of a graph G consists of a set of polygonal obstacles and a drawing of G as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph form a convex or even a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Oksana Firman
creator: Philipp Kindermann
creator: Jonathan Klawitter
creator: Boris Klemz
creator: Felix Klesen
creator: Alexander Wolff
rights: Copyright (c) 2025 Oksana Firman, Philipp Kindermann, Jonathan Klawitter, Boris Klemz, Felix Klesen, Alexander Wolff https://creativecommons.org/licenses/by/4.0
date: 2025-02-10
publication date: 2025-02-10
starting page: 2:1
ending page: 2:21
doi: 10.57717/cgt.v4i1.64

Guest Editors' Foreword
type: item
description: This is the guest editors' foreword for the special issue on selected papers from the 39th European Workshop on Computational Geometry, EuroCG'23.
subject: Original Research Articles
  topic: section
subject: EuroCG2023
  topic: category
creator: Clemens Huemer
creator: Carlos Seara
rights: Copyright (c) 2023 Clemens Huemer, Carlos Seara https://creativecommons.org/licenses/by/4.0
date: 2024-04-30
publication date: 2024-04-30
starting page: 1:1
ending page: 1:1
doi: 10.57717/cgt.v3i2.68

Hosting of the CG:SHOP Challenges
type: item
description: This paper offers insights into the development and hosting of the Computational Geometry: Solving Hard Optimization Problems (CG:SHOP) Challenges. We examine the architecture of the supporting infrastructure, detail the tasks and workloads involved, discuss methods for selecting an effective set of instances, and share the lessons learned from the first seven iterations of the competition.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2023
  topic: category
creator: Phillip Keldenich
creator: Rouven Kniep
creator: Dominik Krupke
rights: Copyright (c) 2025 Dominik Krupke, Phillip Keldenich, Rouven Kniep https://creativecommons.org/licenses/by/4.0
date: 2026-02-19
publication date: 2026-02-19
starting page: 3:1
ending page: 3:25
doi: 10.57717/cgt.v5i4.82

How Packed Is It, Really?
type: item
description: The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve p is c-packed if the length of the curve lying inside any ball is at most c times the radius of the ball, and its congestion is the minimum c for which pi is c-packed. This paper presents a randomized 42-approximation algorithm for computing the congestion of a curve (or any set of segments in the plane). It runs in O(n log2 n) time and succeeds with high probability. Although the approximation factor is large, the running time improves over the previous fastest constant approximation algorithm, which runs in (roughly) O(n4/3) time. We carefully combine new ideas with known techniques to obtain our new near-linear time algorithm.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Sariel Har-Peled
creator: Timothy Zhou
rights: Copyright (c) 2025 Sariel Har-Peled, Timothy Zhou https://creativecommons.org/licenses/by/4.0
date: 2025-06-24
publication date: 2025-06-24
starting page: 6:1
ending page: 6:14
doi: 10.57717/cgt.v4i1.65

On Erdos-Szekeres Maker-Breaker games
type: item
description: The Erdős-Szekeres Maker-Breaker game is a two-player competitive game where both players alternately place points anywhere in the two dimensional Euclidean plane without overlapping, such that no three points are collinear. The first player (Maker) starts the game by placing her point and wants to obtain an empty convex polygon of a given size k such that the vertices of the polygon are chosen from these points and the second player (Breaker) wants to prevent it. We show that Maker wins the game for k <= 8. We also present a winning strategy for Maker for any k in general when Maker is allowed to place (1 + eps) times more points (each round on average) in comparison to Breaker, for any eps > 0. Further, we address the models of the game for equilateral empty convex polygons in the plane and empty convex polygons in square grids.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: CCCG2024
  topic: category
creator: Arun Kumar Das
creator: Tomáš Valla
rights: Copyright (c) 2026 Arun Kumar Das, Tomáš Valla https://creativecommons.org/licenses/by/4.0
date: 2026-02-04
publication date: 2026-02-04
starting page: 3:1
ending page: 3:10
doi: 10.57717/cgt.v5i2.73

An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs
type: item
description: Let V be a set of n points in the plane. The unit-disk graph G = (V, E) has vertex set V and an edge e_uv in E between vertices u, v in V if the Euclidean distance between u and v is at most 1. The weight of each edge e_uv is the Euclidean distance between u and v. Given V and a source point s in V we consider the problem of computing shortest paths in G from s to all other vertices. The previously best algorithm for this problem runs in O(n log2 n) time [Wang and Xue, SoCG'19]. The problem has an Omega(n log n) lower bound under the algebraic decision tree model. In this paper, we present an improved algorithm of O(n log2 n / log log n) time (under the standard real-RAM model). Furthermore, we show that the problem can be solved using O(n log n) comparisons under the algebraic decision tree model, matching the Omega(n log n) lower bound.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: CCCG2024
  topic: category
creator: Bruce Brewer
creator: Haitao Wang
rights: Copyright (c) 2026 Bruce W. Brewer, Haitao Wang https://creativecommons.org/licenses/by/4.0
date: 2026-02-04
publication date: 2026-02-04
starting page: 2:1
ending page: 2:14
doi: 10.57717/cgt.v5i2.69

The Mutual Visibility Problem for Fat Robots with Lights
type: item
description: Given a set of n≥1 unit disk robots in the Euclidean plane, we consider the fundamental problem of providing mutual visibility to them: the robots must reposition themselves to reach a configuration where they all see each other. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between them. This problem was solved by Sharma et al. [ICDCN, 2018] in the luminous robots model, where each robot is equipped with an externally visible light that can assume colors from a fixed set of colors, using 9 colors and O(n) rounds. In this work, we present an algorithm that requires only 2 colors and O(n) rounds. The number of colors is optimal since at least two colors are required even for pointrobots [Di Luna et al., Information and Computation, 2017].
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Rusul J. Alsaedi
creator: Joachim Gudmundsson
creator: André van Renssen
rights: Copyright (c) 2026 Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen https://creativecommons.org/licenses/by/4.0
date: 2026-02-19
publication date: 2026-02-19
starting page: 2:1
ending page: 2:17
doi: 10.57717/cgt.v5i1.81

Quadrangulating a Polygon with a Fixed Spirality using Steiner Points
type: item
description: Let P be an n-sided polygon on the plane, where n >= 4. A quadrangulation of P is a geometric plane graph obtained from P by adding edges to the interior of P so that every finite face is quadrilateral. It is easy to see that $P$ does not always admit a quadrangulation. So Ramaswami et al. introduced "Steiner points", which are auxiliary points added to P which helps quadrangulatability of P. Those points are said to be inner and outer if they are added to the interior and the exterior of P, respectively. They proved that every P with Steiner points added admits a quadrangulation, and estimated the number of those Steiner points by n, when all Steiner points are inner, and when all Steiner points are outer. On the other hand, Hidaka et al. used the notion of "spirality of P" which measures how far P is from being convex, and considered how a polygon P of spirality k admits a quadrangulation with inner Steiner points. In this paper, we consider the quadrangulatability of a polygon of P of spirality k when all Steiner points are outer, and when both inner and outer Steiner points are allowed.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Fumiya Hidaka
creator: Atsuhiro Nakamoto
creator: Daiki Takahashi
rights: Copyright (c) 2025 Fumiya Hidaka, Atsuhiro Nakamoto, Daiki Takahashi https://creativecommons.org/licenses/by/4.0
date: 2025-11-14
publication date: 2025-11-14
starting page: 9:1
ending page: 9:13
doi: 10.57717/cgt.v4i1.59

Capturing the Shape of a Point Set With a Line Segment
type: item
description: Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group's shape carries meaning as well. In this paper, we represent a group's shape using a simple geometric object, a line segment. Specifically, given a radius r, we say a line segment is representative of a point set P of n points if it is within distance r of each point p in P. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius r using the shortest line segment. We describe an algorithm to find the shortest representative segment in O(n log h + h log3 h) time, where h is the size of the convex hull of P. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in P move.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Nathan van Beusekom
creator: Marc van Kreveld
creator: Max van Mulken
creator: Marcel Roeloffzen
creator: Bettina Speckmann
creator: Jules Wulms
rights: Copyright (c) 2025 Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, Jules Wulms https://creativecommons.org/licenses/by/4.0
date: 2025-12-09
publication date: 2025-12-09
starting page: 10:1
ending page: 10:35
doi: 10.57717/cgt.v4i1.74

Approximation Algorithms for Anchored Multiwatchman Routes
type: item
description: We study some variants of the k-Watchman Routes problem, the cooperative version of the classic Watchman Route problem in a simple polygon. The k watchmen may be required to see the whole polygon, or some pre-determined quota of area within the polygon, and we want to minimize the maximum length traveled by any watchman. While the single-watchman (k = 1$ version of the problem has received considerable attention and is relatively well understood, much less is known about the multiple watchmen (k >1) variant(s). We provide the first tight approximability results for the anchored k-Watchman Routes problem in a simple polygon, assuming k is fixed, by a fully-polynomial time approximation scheme. The basis for the FPTAS is provided by an exact dynamic programming algorithm. If k varies (i.e., it is part of the input), we give constant-factor approximations.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Joseph S.B. Mitchell
creator: Linh Nguyen
rights: Copyright (c) 2025 Joseph S.B. Mitchell, Linh Nguyen https://creativecommons.org/licenses/by/4.0
date: 2025-08-03
publication date: 2025-08-03
starting page: 7:1
ending page: 7:17
doi: 10.57717/cgt.v4i1.71

Approximating the Directed Hausdorff Distance
type: item
description: The Hausdorff distance is a metric commonly used to compute the set similarity of geometric sets. For sets containing a total of n points, the exact distance can be computed naively in O(n2) time. In this paper, we show how to preprocess point sets individually so that the Hausdorff distance of any pair can then be approximated in linear time. We assume that the metric is doubling. The preprocessing time for each set is O(n log D) where D is the ratio of the largest to smallest pairwise distances of the input. In theory, this can be reduced to O(n log n) time using a much more complicated algorithm. We compute (1+eps)-approximate Hausdorff distance in (2+1/eps)O(d)n time in a metric space with doubling dimension d. The k-partial Hausdorff distance ignores k outliers to increase stability. Additionally, we give a linear-time algorithm to compute directed k$partial Hausdorff distance for all values of k at once with no change to the preprocessing.
subject: Original Research Articles
  topic: section
subject: CCCG2023
  topic: category
creator: Oliver Chubet
creator: Parth Parikh
creator: Donald Sheehy
creator: Siddharth Sheth
rights: Copyright (c) 2025 Oliver Chubet, Parth Parikh, Donald Sheehy, Siddharth Sheth https://creativecommons.org/licenses/by/4.0
date: 2025-09-08
publication date: 2025-09-08
starting page: 6:1
ending page: 6:16
doi: 10.57717/cgt.v4i2.62

Convex Covering Using Collections of Convex Polygons and Set Cover
type: item
description: In the convex covering problem, we are given a convex polygon with holes P and the goal is to cover P using a small number of convex polygons that lie inside P. In this paper, we solve the problem using the following strategy. We find a big collection of large (often maximal) convex polygons inside P and then solve several set cover problems to find a small subset of the collection that covers the whole polygon. The quality of our heuristics is confirmed by winning the second place in the CG:SHOP 2023 Challenge.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2023
  topic: category
creator: Guilherme da Fonseca
rights: Copyright (c) 2026 Guilherme da Fonseca https://creativecommons.org/licenses/by/4.0
date: 2026-02-19
publication date: 2026-02-19
starting page: 5:1
ending page: 5:10
doi: 10.57717/cgt.v5i4.77

Guest Editors' Foreword
type: item
description: This is the guest editors' foreword for the special issue on the Fifth CG Challenge, CG:SHOP 2023.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
subject: Applications and Engineering
  topic: category
subject: CG:SHOP2023
  topic: category
creator: Sándor P. Fekete
creator: Stefan Schirra
rights: Copyright (c) 2025 Sandor Fekete, Stefan Schirra https://creativecommons.org/licenses/by/4.0
date: 2026-02-19
publication date: 2026-02-19
starting page: 1:1
ending page: 1:1
doi: 10.57717/cgt.v5i4.107

Guest Editors' Foreword
type: item
description: This is the guest editors' foreword for the special issue on selected papers from the 34th Canadian Conference on Computational Geometry (CCCG 2022).
subject: Original Research Articles
  topic: section
subject: CCCG2022
  topic: category
creator: Yeganeh Bahoo
creator: Konstantinos Georgiou
rights: Copyright (c) 2023 Yeganeh Bahoo, Konstantinos Georgiou https://creativecommons.org/licenses/by/4.0
date: 2026-05-18
publication date: 2026-05-18
starting page: 1:1
ending page: 1:1
doi: 10.57717/cgt.v2i2.119

A Linear Time Algorithm for the Flip Distance between Plane Spanning Paths in Convex Point Sets
type: item
description: We provide a linear time algorithm to determine the flip distance between two plane spanning paths on a point set in convex position. At the same time, we show that the happy edge property does not hold in this setting. This has to be seen in contrast to several results for reconfiguration problems where the absence of the happy edge property implies algorithmic hardness of the flip distance problem. Further, we show that our algorithm can be adapted for (1) compatible flips (2) local flips and (3) flips for plane spanning paths in simple polygons.
subject: Original Research Articles
  topic: section
subject: EuroCG2025
  topic: category
creator: Oswin Aichholzer
creator: Joseph Dorfer
rights: Copyright (c) 2026 Oswin Aichholzer, Joseph Dorfer https://creativecommons.org/licenses/by/4.0
date: 2026-02-04
publication date: 2026-02-04
starting page: 2:1
ending page: 2:17
doi: 10.57717/cgt.v5i3.93

Pathwidth of 2-Layer k-Planar Graphs
type: item
description: A bipartite graph G = (X ∪ Y, E) is a 2-layer k-planar graph if it admits a drawing on the plane such that the vertices in X and Y are placed on two parallel lines respectively, edges are drawn as straight-line segments, and every edge involves at most k crossings. Angelini, Da Lozzo, Förster, and Schneck [GD 2020; Comput. J., 2024] showed that every 2-layer k-planar graph has pathwidth at most k + 1. In this paper, we show that this bound is sharp by giving a 2-layer k-planar graph with pathwidth k + 1 for every k ≥ 0. This improves their lower bound of (k + 3) / 2.
subject: Original Research Articles
  topic: section
subject: Computational Geometry
  topic: category
creator: Yuto Okada
rights: Copyright (c) 2026 Yuto Okada https://creativecommons.org/licenses/by/4.0
date: 2026-02-19
publication date: 2026-02-19
starting page: 3:1
ending page: 3:8
doi: 10.57717/cgt.v5i1.101