Cubic Augmentation of Planar Graphs

In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm for the variant of the problem where the input graph has a fixed planar (topological) embedding that has to be preserved by the augmentation. We further generalize this algorithm to test efficiently whether a 3-regular planar augmentation exists that additionally makes the input graph connected or biconnected. If the input graph should become even triconnected, we show that the existence of a 3-regular planar augmentation is again NP-hard to decide.Comment: accepted at ISAAC 201

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)¿k, and we prove that for k=5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time. © 2021 Elsevier B.V

Planar subgraphs without low-degree nodes

We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 1) leads to NP-hard problems.Peer reviewe

Probabilistic solar wind and geomagnetic forecasting using an analogue ensemble or "Similar Day" approach

Effective space-weather prediction and mitigation requires accurate forecasting of near-Earth solar-wind conditions. Numerical magnetohydrodynamic models of the solar wind, driven by remote solar observations, are gaining skill at forecasting the large-scale solar-wind features that give rise to near-Earth variations over days and weeks. There remains a need for accurate short-term (hours to days) solar-wind forecasts, however. In this study we investigate the analogue ensemble (AnEn), or “similar day”, approach that was developed for atmospheric weather forecasting. The central premise of the AnEn is that past variations that are analogous or similar to current conditions can be used to provide a good estimate of future variations. By considering an ensemble of past analogues, the AnEn forecast is inherently probabilistic and provides a measure of the forecast uncertainty. We show that forecasts of solar-wind speed can be improved by considering both speed and density when determining past analogues, whereas forecasts of the out-of-ecliptic magnetic field [ BNBN ] are improved by also considering the in-ecliptic magnetic-field components. In general, the best forecasts are found by considering only the previous 6 – 12 hours of observations. Using these parameters, the AnEn provides a valuable probabilistic forecast for solar-wind speed, density, and in-ecliptic magnetic field over lead times from a few hours to around four days. For BNBN , which is central to space-weather disturbance, the AnEn only provides a valuable forecast out to around six to seven hours. As the inherent predictability of this parameter is low, this is still likely a marked improvement over other forecast methods. We also investigate the use of the AnEn in forecasting geomagnetic indices Dst and Kp. The AnEn provides a valuable probabilistic forecast of both indices out to around four days. We outline a number of future improvements to AnEn forecasts of near-Earth solar-wind and geomagnetic conditions

On euclidean steiner (1 + ϵ)-spanners

Lightness and sparsity are two natural parameters for Euclidean (1 + ϵ)-spanners. Classical results show that, when the dimension d ∈ N and ϵ > 0 are constant, every set S of n points in d-space admits an (1+ϵ)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ϵ > 0 for constant d ∈ N have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1 + ϵ)-spanner. They gave upper bounds of Õ(ϵ-(d+1)/2) for the minimum lightness in dimensions d ≥ 3, and Õ(ϵ-(d-1))/2) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1 + ϵ)-spanners of lightness O(ϵ-1 log Δ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1 + ϵ)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ϵ-d/2) for the lightness and Ω(ϵ-(d-1)/2) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1 + ϵ)-spanners of lightness O(ϵ-1 log n) for n points in Euclidean plane.SCOPUS: cp.pDecretOANoAutActifinfo:eu-repo/semantics/publishe

General position subsets and independent hyperplanes in d-space

Erdős asked what is the maximum number α(n) such that every set of n points in the plane with no four on a line contains α(n) points in general position. We consider variants of this question for d-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed d:Every set H of n hyperplanes in Rd contains a subset S⊆ H of size at least c(nlog n) 1 / d, for some constant c= c(d) > 0 ,such that no cell of the arrangement of H is bounded by hyperplanes of S only.Every set of cqdlog q points in Rd, for some constant c= c(d) > 0 ,contains a subset of q cohyperplanar points or q points in general position.Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math. 2013].SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Diffuse Reflection Radius in a Simple Polygon

It is shown that every simple polygon in general position with n walls can be illuminated from a single point light source s after at most ⌊ (n- 2) / 4 ⌋ diffuse reflections, and this bound is the best possible. A point s with this property can be computed in O(nlog n) time. It is also shown that the minimum number of diffuse reflections needed to illuminate a given simple polygon from a single point can be approximated up to an additive constant in polynomial time.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

The strategic role of corporate online references: building social capital through signaling in business networks

PurposeThis paper aims to conceptualize corporate reference management as a strategic signaling activity in business networks. While research has extensively outlined how firms develop and maintain social capital through business-to-business (B2B) relationships, less is known about how they signal their participation in business networks to develop this social capital. Therefore, this paper conceptualizes B2B references, in particular corporate online references (COR), as a tool through which firms “borrow” attractiveness from their business network. Through the lens of structural social capital theory, COR is shown to capture advantages related to interconnectedness between firms.Design/methodology/approachThe paper reports on a two-step qualitative and quantitative research design. First, the authors undertook a qualitative study that reports on the COR practices of senior business managers. A quantitative study then uses social network analysis (SNA) to audit a digital business network comprising 1,098 firms in a metropolitan area of the UK, referencing to each other through their corporate websites using COR.FindingsThe analyses find that COR practices contribute to building structural social capital in networks through strategic signaling. Firms do so by managing B2B references to craft strategic signals, using five steps: requesting, granting, curating, coding and decoding references. While the existing literature on business marketing portrays reference management as a routine and operational management practice, this investigation conceptualizes reference management, in particular COR, as a strategic activity.Originality/valueTo the best of the authors’ knowledge, this is the first study to use SNA to represent B2B references in the form of COR as a network, which overlaps with (but is not entirely identical to) the business network. Further, the study re-conceptualizes reference management as a strategic signaling activity that leverages the firm’s participation in business networks to build structural social capital by borrowing attractiveness of prestigious business partners that leverages existing structural social capital. Finally, the paper coins and conceptualizes COR as an exemplar of referencing management and offers propositions for further research

Polygons with Prescribed Angles in 2D and 3D

We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A = (α0, …, αn−1), αi ∈ (−π, π), for i ∈ {0, …, n − 1}. The problem of realizing A by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P ⊂ R2 realizing A has at least c crossings, for every c ∈ N, and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P ⊂ R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P ⊂ R3, and for every realizable sequence the algorithm finds a realization. © 2022, Brown University. All rights reserved.Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]