An ortho-unit polygon P is an orthogonal polygon all of whose edges have length one. We study the problem of deciding if two given ortho-unit polygons can be transformed from one into another by performing a sequence of flip operations: replacing an edge whose two endpoints are convex with one whose two endpoints are reflex, and vice versa. We first construct a simpler auxiliary polygon P⁎ related to a given ortho-unit polygon P; any flip applied on P is easily translated to an operation on P⁎. Next, we prove that, if two ortho-unit polygons are flip-equivalent, their corresponding auxiliary polygons must meet some non-trivial similarity criteria, and show that this criteria can be determined in O(n) time for a given pair of ortho-unit polygons of size n. Then, we provide examples of pairs of ortho-unit polygons meeting this criteria but not being flip-equivalent, and give an O(n2)-time algorithm to decide flip equivalence between pairs of ortho-unit polygons. Finally, we show that Ω(n2) flips are sometimes necessary to transform between two flip-equivalent ortho-unit polygons.
In the classical geometric Steiner tree problem, we are given a set of points in the plane and our aim is to find the shortest network interconnecting the set of points. An unlimited number of additional vertices, called Steiner points, may be added to shorten the network. In the minimum k-Steiner tree problem, the number of Steiner points is limited to some nonnegative integer k, which creates additional complexity. This paper improves on the current algorithmic approach to solving the Euclidean k-Steiner tree problem. We introduce a novel pruning test and strengthen existing tests to allow more extensive elimination of sub-optimal topologies during the generation phase of the algorithm. We also introduce a new ILP model for the concatenation phase. Finally, we present experimental results that demonstrate the effectiveness of these novel components. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
A 2-clique in a graph G is a subset S of vertices of G, such that for every two vertices in S, the distance between them in G is at most 2. In this paper, we prove that every 2-clique in unit disk graphs is 3-dominated. This answers an open question posed by Pattillo et al. [10] and provides an improvement of the approximation ratio of their algorithm from 2 to 2 1 3 . (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
A graph is an optimal right angle crossing graph (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 3T-or TTX-framed, that is, the outer face is a triangle that is adjacent to three triangles or to two triangles and a crossing. An optimal pseudo-RAC graph is the topological version of an optimal RAC graph, where the restrictions to straight-line edges and right angle crossings are dropped. 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 n-vertex 3T-and TTX-framed optimal RAC graphs for every n >= 9, and eleven optimal RAC and fourteen optimal pseudo-RAC graphs with at most eight vertices. Optimal pseudo-RAC graphs can be recognized in O(n3) 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. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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 theta is a copy of an axis-aligned histogon rotated by theta in a counterclockwise direction. Our goal is to compute a largest inscribed histogon and a smallest circumscribed histogon over all orientations in [0, pi). 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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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 (log n)O (1) multiplicative factors. It is also shown that the same approach can be generalised to obtain efficient approximation algorithms for computing an optimal weight rho-subgraphs where rho is a suitable hereditary property. Applying our paradigm, we establish, for every k > 2 and p is an element of[1, infinity], that MWIS of the intersection graph of a given collection of weighted k-dimensional Lp spheres (having a common radius) can be efficiently approximated within a multiplicative factor of (log n)(k-1). The running time can be brought down to O (n(logn)) at the cost of increasing the approximation guarantee to c(k,p)(logn)(k-1), for some constant c(p,k) depending only on p and k. 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 k-dimensional, L-p-spaces, for every k >2 and p is an element of(0, infinity]. In a related development, we also establish the following graph theoretic result which will be of independent interest: For every p is an element of[2, infinity] and for every G, there is a k > 1 such that G is isomorphic to the IG of a collection of k-dimensional Lp-spheres of a common radius. The minimum value of such a k is referred to as the p-sphericity of G. Also, applying our paradigm, one obtains for every k > 2, an efficient algorithm which, given a collection B of weighted k-dimensional axis-parallel boxes, finds a (log n)(k-1)-approximation to MWIS. For the unweighted case, the running time can be improved to O (n(logn)(2)). (c) 2025 Published by Elsevier B.V.
We study the problem of multiple anchored watchman routes, where we are given m 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,2 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 O (m log m + n log n) time, using O (m + n) storage, and the min-sum algorithm takes O (n2 log m + m log m) time, also using O (m + n) storage. For generalized Minbar polygons, we prove NP-hardness for the min-sum and min-max criteria, and present approximation algorithms for both criteria: an O (log(m + n))-approximation taking O(m4n2) time for the min-sum criterion, and a (pi + 3)-approximation taking O(m3n2) time for the min-max criterion. 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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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 pi 2 . The necessary and sufficient condition for a subcubic plane graph to have a rectangular drawing was already given by Rahman et al. [14]. This result was extended to a drawing of graphs on an annulus by Hasheminezhad et al. [5]. 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 M & ouml;bius band. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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. 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.
In this work, we present VHS (Voxelized Homological Simplification), 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 https://github.com/ davidletscher/VHS/. (c) 2025 Published by Elsevier B.V.
Given a set P of n points in R2, in general position, denote by N2(P) the number of empty triangles with vertices in P. In this paper we investigate by how much N2(P) changes if a point xis removed from P. By constructing a graph G2(x, P) based on the arrangement of the empty triangles incident on x, we transform this geometric problem to the problem of counting triangles in the graph G2(x, P). We study properties of the graph G2(x, P) and, in particular, show that it is diamond-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemer & eacute;di problem. We also derive similar bounds for the growth rate of the number of empty simplices for point sets in Rd. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Given a set P of n points in the plane, the unit-disk graph G(P)is a graph with Pas its vertex set such that two points of P have an edge if their Euclidean distance is at most 1. We consider the problem of computing a maximum clique in G(P). The previously best algorithm for the problem runs in O(n7/3+o(1)) time. We show that the problem can be solved in O(n log n+nK4/3+o(1)) time, where K is the maximum clique size. The algorithm is faster than the previous one when K = o(n). In addition, if Pis in convex position, we give a randomized algorithm that runs in O(n15/7+o(1)) = O(n2.143) worst-case time and the algorithm can compute a maximum clique with high probability. For points in convex position, one special case we solve is when a point in the maximum clique is given; we present an O(n2log n) time (deterministic) algorithm for this special case. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Let alpha(Fqd,p) be the maximum possible size of a point set in general position in the prandom subset of Fqd. In this note, we determine the order of magnitude of alpha (Fq3, p) up to a polylogarithmic factor by proving a balanced supersaturation result for the sets of 4 points in the same plane. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let G be a complete edge-weighted graph on n vertices. To each subset of vertices of G assign the cost of the minimum spanning tree of the subset as its weight. Suppose that n is a multiple of some fixed positive integer k. The k-matching problem is the problem of finding a partition of the vertices of G into k-sets (sets of k elements), that minimizes the sum of the weights of the k-sets. The case of k = 3 has been shown to be NP-hard [Johnsson et al., 1998]. In the Euclidean version, the vertices of G are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean k-matching problem. We show that, for every fixed k >= 3, the Euclidean k-matching problem is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case of k = 3. We also show that the problem remains NP-hard if the trees are required to be paths. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Given a set P of n points in the plane, the two-line center problem asks to find two lines that minimize the maximum distance from each point in P to its closer one of the two resulting lines. The currently best algorithm for the problem takes O (n2 log2 n) time by Jaromczyk and Kowaluk in 1995. In this paper, we present faster algorithms for three variants of the two-line center problem in which the orientations of the resulting lines are constrained. Specifically, our algorithms solve the problem in O(n log n) time when the orientations of both lines are fixed; in O(nlog3n) time when the orientation of one line is fixed; and in O(n2 alpha(n) log n) time when the angle between the two lines is fixed, where alpha(n) denotes the inverse Ackermann function. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this note, we show that extremal Szemer & eacute;di-Trotter configurations are rigid in the following sense: If P,L are sets of points and lines determining at least C|P|2/3|L|2/3 incidences, then there exists a collection P ' of points of size at most k = k0(C) 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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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 a time complexity of $n^{d+\mathcal{O}(1)}\text{poly}(n,d)$. Additionally, we present a randomized algorithm with a complexity of $2^{\mathcal{O}(r(d-r))} \times \text{poly}(n, d)$. This algorithm samples subspaces characterized in terms of a Grassmannian manifold. By employing such sampling method, we ensure a high likelihood of capturing the optimal subspace, with the success probability $(1 - \delta)^T$. Where $\delta$ represents the probability that a sampled subspace does not contain the optimal solution, and $T$ is the number of subspaces sampled, proportional to $2^{r(d-r)}$. Our use of higher-degree Voronoi diagrams and Grassmannian based sampling offers a clearer conceptual pathway and practical advantages, particularly in handling large datasets or higher-dimensional settings.
For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of n points in ℝ^d , there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least logn/(4d) . Apart from the 1/(4d) factor, this bound is the best possible. As for the abstract setting, we show that for every n-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree (√(logn/loglogn)) .
We study two problems related to the city guarding and the art gallery problems. 1. Given a city with k rectangular buildings, we prove that 3k + 1 cameras of 180 degrees 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) [7]. 2. Given k orthogonally convex polygons of total m vertices in the plane, we prove that m2 + k + 1 cameras of 180 degrees field of view are always sufficient to guard the free space (avoiding all the polygons). This answers another conjecture of Daescu and Malik (2021) [8]. 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. We then generalize the above bounds to arbitrary convex-shape buildings. We can guard the free space of k buildings of total size m by m-k + 1 cameras. For k simple polygons with c convex vertices in the plane we can guard the free space by c-k + 1 cameras. Again, both these bounds are tight. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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 n >= r be positive integers. The graph K-n (R), is the complete balanced r-partite graph on n vertices, in which every set of the partition has at least & LeftFloor;n/r & RightFloor; vertices. The balanced layered graph, L-n (R), is an r-partite graph on n vertices, where n is multiple of r. Every partition of L-n (R) contains n/r vertices; for every 1 <= i <= r-1, all the vertices in the i-th partition are adjacent to all the vertices in the (i+1)-th partition, and these are the only edges of L-n (R). In this paper, we give upper bounds on the rectilinear crossing numbers of K-n (R) and L-n (R).
