Embedding a graph in the grid of a surface with the minimum number of bends is NP-hard

This paper is devoted to the study of graph embeddings in the grid of non-planar surfaces. We provide an adequate model for those embeddings and we study the complexity of minimizing the number of bends. In particular, we prove that testing whether a graph admits a rectilinear (without bends) embedding essentially equivalent to a given embedding, and that given a graph, testing if there exists a surface such that the graph admits a rectilinear embedding in that surface are NP-complete problems and hence the corresponding optimization problems are NP-hard

Single bend wiring on surfaces

The following problem of rectilinear routing is studied: given pairs of points on a surface and a set of permissible orthogonal paths joining them, whether is it possible to choose a path for each pair avoiding all intersections. We prove that if each pair has one or two possible paths to join it, then the problem is solvable in quadratic time, and otherwise it is NP-complete. From that result, we will obtain that the problem of finding a surface of minimum genus on which the wires can be laid out with only one bend is NP-hard

Time Series on Functional Service Life of Buildings using Fuzzy Delphi Method

The functional service life of heritage buildings, defined as the time period during which the building fulfils the requirements for which it was designed, is a complex system that has still not been fully resolved and continues to be the object of research regarding its social, economic and cultural importance. This paper presents an application for analysing time series that reflect the state of building performance over time. To this end, historical time records are used that provided data that could be interpreted by experts in the field. The latter can then evaluate the input variables (vulnerability and risk) using the expert system for predicting the service life of buildings, Fuzzy Building Service Life (FBSL), this methodology put together the fuzzy logic tools and Delphi method. This model provides output data on the state of functionality or performance of each buildings at each moment in time whenever information records are available. The Delphi Method is used to eliminate expert subjectivity, establishing an FDM-type assessment methodology that effectively quantifies the service life of buildings over time. The application is able to provide significant data when generating future preventive maintenance programmes in architectural-cultural heritage buildings. It can also be used to optimise the resources invested in the conservation of heritage buildings. In order to validate this system, the FDM methodology is applied to some specific building examples.Ministerio de Economía y Competitividad de España, Project ART-RISK - BIA2015-64878-RMinisterio de Economía y Competitividad de España MTM 2015-65397-

Labeling Subway Lines

Graphical features on map, charts, diagrams and graph drawings usually must be annotated with text labels in order to convey their meaning. In this paper we focus on a problem that arises when labeling schematized maps, e.g. for subway networks. We present algorithms for labeling points on a line with axis-parallel rectangular labels of equal height. Our aim is to maximize label size under the constraint that all points must be labeled. Even a seemingly strong simplification of the general point-labeling problem, namely to decide whether a set of points on a horizontal line can be labeled with sliding rectangular labels, turns out to be weakly NPcomplete. This is the first labeling problem that is known to belong to this class. We give a pseudo-polynomial time algorithm for it. In case of a sloping line points can be labeled with maximum-size square labels in O(n log n) time if four label positions per point are allowed and in O(n 3 log n) time if labels can slide. We also investigate rectangular labels

Reporting Bichromatic Segment Intersections from Point Sets

In this paper, we introduce a natural variation of the problem of computing all bichromatic intersections between two sets of segments. Given two sets R and B of n points in the plane defining two sets of segments, say red and blue, we present an O(n2) time and space algorithm for solving the problem of reporting the set of segments of each color intersected by segments of the other color. We also prove that this problem is 3-Sum hard and provide some illustrative examples of several point configurations

Preventive Conservation and Restoration Monitoring of Heritage Buildings Based on Fuzzy Logic

This article discusses the usability of the Art-Risk 3.0 software for research on the conservation of heritage buildings. It is a new and free software based on fuzzy logic, which enables the assessment of preventive conservation and surveillance of the restoration of heritage buildings over a period of time. This artificial intelligence-based tool considers the vulnerability of buildings, their environ ments, and their management to evaluate the necessity of their restoration or preventive con servation. To validate the Art-Risk 3.0, 500 theoretical case studies were analyzed, and a 14th century Mudejar-Gothic-style Church in Seville, Spain was studied both before and after its restora tion to identify post-restoration changes. This proof of concept demonstrates the capability of the Art-Risk 3.0 software to analyze environmental impacts on the vulnerability, risk, and functional service life of buildings, and assess the effectiveness of restoration activities. Additionally, this software identifies the most problematic factors and the necessity of restoration.Ministerio de Economía y Competitividad BIA2015- 64878-R (RETOS)Ministerio de Ciencia e Innovación PID2019-107257RB-I00 (FENIX)Ministerio de Ciencia, Innovación y Universidades EQC2019-005780-P (Ambulab-LAB)Junta de Andalucía PYC20 RE 034 UPO RESILIENT-TOURISMMinisterio de Ciencia e Innovación PTA2019-01688

Monochromatic geometric k-factors for bicolored point sets with auxiliary points

Given a bicolored point set S, it is not always possible to construct a monochromatic geometric planar k-factor of S. We consider the problem of finding such a k-factor of S by using auxiliary points. Two types are considered: white points whose position is fixed, and Steiner points which have no fixed position. Our approach provides algorithms for constructing those k-factors, and gives bounds on the number of auxiliary points needed to draw a monochromatic geometric planar k-factor of S

Monochromatic geometric k-factors in red-blue sets with white and Steiner points

We study the existence of monochromatic planar geometric k-factors on sets of red and blue points. When it is not possible to find a k-factor we make use of auxiliary points: white points, whose position is given as a datum and which color is free; and Steiner points whose position and color is free. We present bounds on the number of white and/or Steiner points necessary and/or sufficient to draw a monochromatic planar geometric k-factor

Cover contact graphs

We study problems that arise in the context of covering certain geometric objects called seeds (e.g., points or disks) by a set of other geometric objects called cover (e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, respectively, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in three types of tasks, both in the general case and in the special case of seeds on a line: (a) deciding whether a given seed set has a connected CCG, (b) deciding whether a given graph has a realization as a CCG on a given seed set, and (c) bounding the sizes of certain classes of CCG’s. Concerning (a) we give efficient algorithms for the case that seeds are points and show that the problem becomes hard if seeds and covers are disks. Concerning (b) we show that this problem is hard even for point seeds and disk covers (given a fixed correspondence between graph vertices and seeds). Concerning (c) we obtain upper and lower bounds on the number of CCG’s for point seeds

Cover Contact Graphs

Es una ponencia presentada al 15th International Symposium on Graph Drawing (2007)We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).German Research Foundation WO 758/4-