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.
