Edge Intersection Research Articles

New complexity measures reveal that topographic loops of human alpha phase potentials are more complex in drowsy than in wake.

A number of measures, stemming from nonlinear dynamics, exist to estimate complexity of biomedical objects. In most cases they are appropriate, but sometimes unconventional measures, more suited for specific objects, are needed to perform the task. In our present work, we propose three new complexity measures to quantify complexity of topographic closed loops of alpha carrier frequency phase potentials (CFPP) of healthy humans in wake and drowsy states. EEG of ten adult individuals was recorded in both states, using a 14-channel montage. For each subject and each state, a topographic loop (circular directed graph) was constructed according to CFPP values. Circular complexity measure was obtained by summing angles which directed graph edges (arrows) form with the topographic center. Longitudinal complexity was defined as the sum of all arrow lengths, while intersecting complexity was introduced by counting the number of intersections of graph edges. Wilcoxon's signed-ranks test was used on the sets of these three measures, as well as on fractal dimension values of some loop properties, to test differences between loops obtained in wake vs. drowsy. While fractal dimension values were not significantly different, longitudinal and intersecting complexities, as well as anticlockwise circularity, were significantly increased in drowsy. Graphical abstract An example of closed topographic carrier frequency phase potential (CFPP) loops, recorded in one of the subjects in the wake (A) and drowsy (C) states. Lengths of loop graph edges, r(c j, c j+1), plotted against the series of EEG channels with decreasing CFPP values, c j , in the wake (B) and drowsy (D) states. Conventional fractal analysis did not reveal any difference between them; therefore, three new complexity measures were introduced.

On the Number of Edges of a Uniform Hypergraph with a Range of Allowed Intersections

We study the quantity p(n, k, t1, t2) equal to the maximum number of edges in a k-uniform hypergraph having the property that all cardinalities of pairwise intersections of edges lie in the interval [t1, t2]. We present previously known upper and lower bounds on this quantity and analyze their interrelations. We obtain new bounds on p(n, k, t1, t2) and consider their possible applications in combinatorial geometry problems. For some values of the parameters we explicitly evaluate the quantity in question. We also give a new bound on the size of a constant-weight error-correcting code.

Comparison of Three Theoretical Methods for Determining Dry and Wet Edges of the LST/FVC Space: Revisit of Method Physics

Land surface temperature and fractional vegetation coverage (LST/FVC) space is a classical model for estimating evapotranspiration, soil moisture, and drought monitoring based on remote sensing. One of the key issues in its utilization is to determine its boundaries, i.e., the dry and wet edges. In this study, we revisited and compared three methods that were presented by Moran et al. (1994), Long et al. (2012), and Sun (2016) for calculating the dry and wet edges theoretically. Results demonstrated that: (1) for the dry edge, the Sun method is equal to the Long method and they have greater vegetation temperature than that of the Moran method. (2) With respect to the wet edge, there are greater differences among the three methods. Generally, Long’s wet edge is a horizontal line equaling air temperature. Sun’s wet edge is an oblique line and is higher than that of the Long’s. Moran’s wet edge intersects them with a higher soil temperature and a lower vegetation temperature. (3) The Sun and Long methods are simpler in calculation and can circumvent some complex parameters as compared with the Moran method. Moreover, they outperformed the Moran method in a comparison of estimating latent heat flux (LE), where determination coefficients varied between 0.45 ~ 0.66 (Sun), 0.47 ~ 0.68 (Long), and 0.39 ~ 0.57 (Moran) among field stations.

On edge intersection graphs of paths with 2 bends

An EPG-representation of a graph G is a collection of paths in the plane square grid, each corresponding to a single vertex of G, so that two vertices are adjacent if and only if their corresponding paths share infinitely many points. In this paper we focus on graphs admitting EPG-representations by paths with at most 2 bends. We show hardness of the recognition problem for this class of graphs, along with some subclasses.We also initiate the study of graphs representable by unaligned polylines, and by polylines, whose every segment is parallel to one of prescribed slopes. We show hardness of recognition and explore the trade-off between the number of bends and the number of slopes.

Studying Properties of Composite Nano-PAA-Au Array for the Optimal SERS Sensitivity

The composite structure of nano-porous anodic alumina (nano-PAA) coated with noble metal layer has been demonstrated to be one kind of high-sensitive surface-enhanced Raman scattering (SERS)-active substrate. However, the experimental investigations on influences of parameters were restricted by correlation between variables. And in relevant simulation studies, nano-tips at intersections of unit edges on the surface and the metal attaching to inside walls of nano-holes which are crucial for the enhancement of electric field were commonly idealized. To obtain the optimal structure of nano-PAA-Au array for SERS applications, numerical evaluations of the electric field distributions and intensities at hotspots were carried out on a proposed precise model. The influence of structure and excitation parameters which determined the electromagnetic enhancement have been systematically investigated. The numerical data shows that the intensity and distribution of surface electric field is evidently affected by both the characteristics of nano-tips and the metal attaching to the inside wall of holes. In addition, the designed substrate was synthesized accordingly and demonstrated to be higher sensitive with an achieved enhancement factor of 1.04 × 107. The study would be helpful in designing high-sensitive SERS-active substrates.

Clique coloring [formula omitted]-EPG graphs

We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.

Hanani-Tutte for Radial Planarity

A drawing of a graph $G$ is radial if the vertices of $G$ are placed on concentric circles $C_1, \ldots, C_k$ with common center $c$, and edges are drawn radially: every edge intersects every circle centered at $c$ at most once. $G$ is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of $G$ are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph $G$ is radial planar if $G$ has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.

A Discrete Fruit Fly Optimization Algorithm for the Traveling Salesman Problem.

The fruit fly optimization algorithm (FOA) is a newly developed bio-inspired algorithm. The continuous variant version of FOA has been proven to be a powerful evolutionary approach to determining the optima of a numerical function on a continuous definition domain. In this study, a discrete FOA (DFOA) is developed and applied to the traveling salesman problem (TSP), a common combinatorial problem. In the DFOA, the TSP tour is represented by an ordering of city indices, and the bio-inspired meta-heuristic search processes are executed with two elaborately designed main procedures: the smelling and tasting processes. In the smelling process, an effective crossover operator is used by the fruit fly group to search for the neighbors of the best-known swarm location. During the tasting process, an edge intersection elimination (EXE) operator is designed to improve the neighbors of the non-optimum food location in order to enhance the exploration performance of the DFOA. In addition, benchmark instances from the TSPLIB are classified in order to test the searching ability of the proposed algorithm. Furthermore, the effectiveness of the proposed DFOA is compared to that of other meta-heuristic algorithms. The results indicate that the proposed DFOA can be effectively used to solve TSPs, especially large-scale problems.

Revealing Patterns and Trends of Mass Mobility Through Spatial and Temporal Abstraction of Origin-Destination Movement Data.

Origin-destination (OD) movement data describe moves or trips between spatial locations by specifying the origins, destinations, start, and end times, but not the routes travelled. For studying the spatio-temporal patterns and trends of mass mobility, individual OD moves of many people are aggregated into flows (collective moves) by time intervals. Time-variant flow data pose two difficult challenges for visualization and analysis. First, flows may connect arbitrary locations (not only neighbors), thus making a graph with numerous edge intersections, which is hard to visualize in a comprehensible way. Even a single spatial situation consisting of flows in one time step is hard to explore. The second challenge is the need to analyze long time series consisting of numerous spatial situations. We present an approach facilitating exploration of long-term flow data by means of spatial and temporal abstraction. It involves a special way of data aggregation, which allows representing spatial situations by diagram maps instead of flow maps, thus reducing the intersections and occlusions pertaining to flow maps. The aggregated data are used for clustering of time intervals by similarity of the spatial situations. Temporal and spatial displays of the clustering results facilitate the discovery of periodic patterns and longer-term trends in the mass mobility behavior.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

Development of the interfacial air layer in the non-aerated region of high-velocity spillway flows. Instabilities growth, entrapped air and influence on the self-aeration onset

Self-aeration is traditionally explained by the water turbulent boundary layer outer edge intersection with the free surface. This paper presents a discussion on the commonly accepted hypothesis behind the computation of the critical point of self-aeration in spillway flows and a new formulation is proposed based on the existence of a developing air flow over the free surface. Upstream of the inception point of self-aeration, some surface roughening has been often reported in previous studies which consequently implies some entrapped air transport and air–water flows coupling. Such air flow is proven in this study by presenting measured air velocities and computing the air boundary layer thickness for a 1V:2H smooth chute flow. Additionally, the growth rate of free surface waves has been analysed by means of Ultrasonic Sensors measurements, obtaining also the entrapped air concentration. High-speed camera imaging has been used for qualitative study of the flow perturbations.

On Graphs, Which Can be Drawn on an Orientable Surface with Small Number of Intersections on an Edge

Let k and g be nonnegative integers. A graph is said to be k-nearly g-spherical if it can be drawn on an orientable surface of genus g so that each edge intersects at most k other edges in interior points. It is proved that if k ≤ 4, then the edge number of a k-nearly g-spherical graph on v vertices does not exceed (k + 3)(v + 2g − 2). It is also shown that the chromatic number of a k-nearly g-spherical graph does not exceed $$ \frac{2k+7+\sqrt{4{k}^2+12k+1+16\left(k+3\right)g}}{2} $$ . Bibliography: 4 titles.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern proved that every graph can be represented as the edge intersection graph of paths on a grid, i.e., one can associate to each vertex of the graph a nontrivial path on a grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is B4-EPG, by embedding the circle into a rectangle of the grid. In this paper we prove that every circular-arc graph is B3-EPG, but if we restrict ourselves to rectangular representations there exist some graphs that require paths with four bends. We also show that normal circular-arc graphs admit rectangular representations with at most two bends per path. Moreover, we characterize graphs admitting a rectangular representation with at most one bend per path by forbidden induced subgraphs, and we show that they are a subclass of normal Helly circular-arc graphs.

SUBSTANTIATION OF SURGICAL APPROACH IN IATROGENIC INJURIES OF THE BILE-EXCRETING DUCTS

The results of examination and treatment were analyzed in 51 patients with iatrogenic injuries of the bile-excreting ducts. Patients were divided into 5 groups according to international classification (EAES, 2013). It depended on the time of detection, the nature and scale of damage of the bile ducts, mechanism of injury, development of infectious and septic complications. Injuries of the main bile duct were detected intraoperatively (n = 14). The complete intersection was in 10 patients (the first group) and the edge intersection--in 4 cases (the second group). Iatrogenic injuries of the bile-excreting ducts were revealed in 37 patients in postoperative period. There were the complete intersections in 28 cases (the third group) and the edge intersections--in 7 cases (the fifth group). Injuries of additional bile ducts were determined in 2 patients (the fifth group). An analysis of the main qualifying features of iatrogenic injuries of the bile-excreting ducts allowed defining indications to reconstructive-restorative surgery in 60.8% patients, restorative operations--in 29.4%, an external drainage--in 5.8% and reclipping of additional bile ducts in relaparoscopy--in 3.9%. The rational surgical approach allowed obtaining perfect results in 65.8% and good, satisfactory results in immediate and long-term period with low postoperative lethality of 1.95%. The study of diagnostics results and treatment of the patients with iatrogenic injuries of the bile- excreting ducts indicated about reasonability of assessment of main factors, which are based on iatrogenic injuries according to the EAES classification. An individual program of examination and more rational variant of surgery could be chosen due to this approach, which provides minimization of negative results and good quality of life.

Graphs of edge-intersecting and non-splitting paths

The families of Edge Intersection Graphs of Paths in a tree (resp. in a grid) EPT (resp. EPG) are well studied graph classes. Recently we introduced the class of graphs of Edge-Intersecting and Non-Splitting Paths in a Tree (ENPT) [5]. In this model, two vertices are adjacent if they represent two intersecting paths of a tree whose union is also a path. In this study we generalize this graph class by allowing the host graph to be an arbitrary graph. These are the graphs of Edge-Intersecting and Non-Splitting Paths ENP. Although the Edge Intersection Graphs of Paths in an arbitrary graph includes all graphs, we show that this is not the case for ENP. We also show that the class ENP coincides with the family of graphs of Edge-Intersecting and Non-Splitting Paths in a Grid (ENPG). Following similar studies for EPG graph class, we study the implications of restricting the number of bends of the individual paths in the grid. We show that such a restriction restricts the graph class. Specifically, by restricting the number of bends one gets an infinite sequence of classes such that every class is properly included in the next one.

Strong cliques and equistability of EPT graphs

In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.

Graphs of edge-intersecting non-splitting paths in a tree: Representations of holes—Part I

Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P of the tree T, and Ept(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices are the vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path).In this work, motivated by an application in all-optical networks, we define the graph Enpt(P) of edge-intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in P, and an edge between every pair of paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=Enpt(P), and we say that 〈T,P〉 is a representation of G. We first show that cycles, trees and complete graphs are ENPT graphs.Our work follows the lines of Golumbic and Jamison’s research (Golumbic and Jamison, 1985) in which they defined the EPT graph class, and characterized the representations of chordless cycles (holes). It turns out that ENPT holes have a more complex structure than EPT holes. In our analysis, we assume that the EPT graph corresponding to a representation of an ENPT hole is given. We also introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In this Part I, using the results of Golumbic and Jamison as building blocks, we characterize (a) EPT, ENPT pairs that satisfy (P1)–(P3), and (b) the unique minimal representation of such pairs.

‘Mathematical’ Cracks Versus Artificial Slits: Implications in the Determination of Fracture Toughness

An analytic solution is introduced for the stress field developed in a circular finite disc weakened by a central slit of arbitrary ratio of its edges and slightly rounded corners. The disc is loaded by radial pressure applied along two finite arcs of its periphery, anti-symmetric with respect to the disc’s center. The motive of the study is to consider the stress field in a disc with a mechanically machined slit (finite distance between the two lips) in juxtaposition to the respective field in the same disc with a ‘mathematical’ crack (zero distance between lips), which is the configuration adopted in case the fracture toughness of brittle materials is determined according to the standardized cracked Brazilian-disc test. The solution is obtained using Muskhelishvili’s complex potentials’ technique adopting a suitable conformal mapping function found, also, in Savin’s milestone book. For the task to be accomplished, an auxiliary problem is first solved, namely, the infinite plate with a rectangular slit (in case the resultant force on the slit is zero and also the stresses and rotations at infinity are zero), by mapping conformally the area outside the slit onto the mathematical plane with a unit hole. The formulae obtained for the complex potentials permit the analytic exploration of the stress field along some loci of crucial practical importance. The influence of the slit’s width on the local stress amplification and also on the stress concentration around the crown of the slit is quantitatively described. In addition, the role of the load-application mode (compression along the slit’s longitudinal symmetry axis and tension normal to it) is explored. Results indicate that the two configurations are not equivalent in terms of the stress concentration factor. In addition, depending on the combination of the slit’s width and the load-application mode, the point where the normal stress along the slit’s boundary is maximized ‘oscillates’ between the central point of the slit’s short edge (intersection of the slit’s longitudinal axis with its perimeter) and the slit’s corners.

Visual positioning of rectangular lead components based on Harris corners and Zernike moments

With the increasing necessities for reliable printed circuit board (PCB) product, there has been a considerable demand for high speed and high precision vision positioning system. To locate a rectangular lead component with high accuracy and reliability, a new visual positioning method was introduced. Considering the limitations of Ghosal sub-pixel edge detection algorithm, an improved algorithm was proposed, in which Harris corner features were used to coarsely detect the edge points and Zernike moments were adopted to accurately detect the edge points. Besides, two formulas were developed to determine the edge intersections whose sub-pixel coordinates were calculated with bilinear interpolation and conjugate gradient method. The last experimental results show that the proposed method can detect the deflection and offset, and the detection errors are less than 0.04° and 0.02 pixels.

Edge intersection graphs of [formula omitted]-shaped paths in grids

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞,⌜,⌟,⌝, and we consider zero bend paths (i.e., ∣ and–) to be degenerate ⌞’s. These graphs, called B1-EPG graphs, were first introduced by Golumbic et al. (2009). We consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {⌞},{⌞,⌜},{⌞,⌝}, and {⌞,⌜,⌝}) and we denote the classes by [⌞],[⌞,⌜],[⌞,⌝], and [⌞,⌜,⌝] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [⌞]⊊[⌞,⌜],[⌞,⌝]⊊[⌞,⌜,⌝]⊊B1-EPG and [⌞,⌜] is incomparable with [⌞,⌝]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩[⌞].