Pathwidth Research Articles

Gallai's path decomposition conjecture for graphs with treewidth at most 3

AbstractA path decomposition of a graph is a set of edge‐disjoint paths of that covers the edge set of . Gallai (1968) conjectured that every connected graph with vertices admits a path decomposition of size at most . Gallai's conjecture was verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex of odd degree. Recently, Bonamy and Perrett verified Gallai's conjecture for graphs with maximum degree at most 5. In this paper, we verify Gallai's conjecture for graphs with treewidth at most 3. Moreover, we show that the only graphs with treewidth at most 3 that do not admit a path decomposition of size at most are isomorphic to or , the graph obtained from by removing an edge.

On Computing the Path Number of a Graph

Gallai (1966) conjectured that the edge set of every graph G on n vertices can be covered by at most ⌈n/2⌉ edge-disjoint paths. Such a covering by edge-disjoint paths is called a path decomposition, and the size of a path decomposition with a minimum number of elements is called the path number of G. Peroche (1984) proved that the problem of computing the path number is NP-Complete; and Constantinou and Ellinas (2018) proved that it is polynomial for a family of complete bipartite graphs. In this paper we present an Integer Linear Programming model for computing the path number of a graph. This allowed us to verify Gallai's Conjecture for a large collection of graphs. As a result, following a work of Heinrich, Natale and Streicher on cycle decompositions (2017), we verify Gallai's Conjecture for graphs with at most 11 vertices; for bipartite graphs with at most 16 vertices; and for regular graphs with at most 14 vertices.

A study on the pendant number of graph products

Abstract A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

Continuous Monotonic Decomposition of Some Standard Graphs by using an Algorithm

In this paper we elaborate an algorithm to compute the necessary and sufficient conditions for the continuous monotonic star decomposition of the bipartite graph Km,r and the number of vertices and the number of disjoint sets. Also an algorithm to find the tensor product of Pn  Ps has continuous monotonic path decomposition. Finally we conclude that in this paper the results described above are complete bipartite graphs that accept Continuous monotonic star decomposition. There are many other classes of complete tripartite graphs that accept Continuous monotonic star decomposition. In this research article Extended to complete m-partite graphs for grater values of m. Also the algorithm can be developed for the tensor product of different classes such as Cn Wn K1,n , , with Pn

Spatial Organization of Human Population and Wastewater Treatment Plants in Urbanized River Basins

AbstractDischarge from multiple wastewater treatment plants (WWTPs) distributed in urbanized river basins contributes to impairments of river water‐quality and aquatic ecosystem integrity, with size and location of WWTPs determined by population distribution within a river basin. Here we used geo‐referenced data for WWTPs in Germany to investigate the spatial organization of three attributes of interest in this study: population, population equivalents (the aggregated population served by each WWTP), and the number/sizes of WWTPs. To this end, we selected as case studies three large urbanized river basins (Weser, Elbe, and Rhine), home to about 70% of the population in Germany. We employed fractal river networks as structural platforms to examine the spatial patterns from two perspectives: spatial hierarchy (stream order) and patterns along longitudinal flow paths (width function). Moreover, we proposed three dimensionless scaling indices to quantify (1) human settlement preferences by stream order, (2) non‐sanitary flow contribution to total wastewater treated at WWTPs, and (3) degree of centralization in WWTPs locations. Across the three river basins, we found scale‐invariant distributions for each of the three attributes with stream order, quantified using extended Horton scaling ratios. We found a weak downstream clustering of population in the three basins. Variations in population equivalent clustering among different class‐sizes of WWTPs reflected the size, number, and locations of urban agglomerations in these river basins. We discussed the applicability of this approach to other large urbanized basins to analyze spatial organization of population and WWTPs.

Scenarios of crashes involving light mopeds on urban bicycle paths

In the Netherlands, cyclists have to share the bicycle path with light moped riders. These riders are allowed to ride 25 km/h and do not have to wear a helmet (Dutch regulation). Due to several trends such as traffic congestion and the introduction of the scooter model, light mopeds have become more popular, both among older and younger people. This has led to an increased traffic density on bicycle paths as well as concerns about the safety of cyclists. In response to these concerns, several Dutch cities would like to ban light moped riders (LMRs) from the bicycle path and let them ride on the carriageway. However, it is uncertain what the consequences would be for the safety of light moped riders. Moreover, it is not clear to what extent the shared use of bicycle paths leads to serious crashes between cyclists and LMRs. Therefore, an in-depth crash investigation study was carried out to gain more insight into the factors and circumstances that influence the occurrence and consequences of light moped crashes on bicycle paths.A dedicated team for in-depth road crash investigation collected and analyzed detailed information on 36 light moped crashes that occurred on an urban bicycle path. This resulted in a description of the course of events for every crash that was analyzed, including a list of factors that contributed to the occurrence of the crash and possible injuries. Crashes with a similar course of events and a comparable combination of contributory factors were grouped into (sub)types of light moped crashes.Six types of crashes were identified. Based on the contributory crash factors of the identified crash types, remedial measures can be developed to prevent similar crashes from occurring in the future. Moving the LMR to the carriageway is only advisable on 30 km/h roads. Alternative measures to improve the safety of both cyclists and light moped riders include: 1) removing obstacles such as poles from the bicycle path, 2) following guidelines on the minimum width of bicycle paths given traffic volumes, 3) improving sight distances at intersections, 4) traffic light control without conflicts between traffic flows, and 5) introducing a helmet law for light moped riders and their passengers.

Childhood adversity and adolescent psychopathology: evidence for mediation in a national longitudinal cohort study.

Childhood adversity is a well-established risk factor for psychopathology; however, many who experience adversity do not go on to develop psychopathology. Poor self-concept and poor parental support are known risk factors for adolescent psychopathology, which may account for some of this mechanism. To investigate candidate mediators in the relationship between childhood adversity and psychopathology. We used data from the age 9 and 13 waves of the child-cohort of the Growing Up in Ireland study. We undertook mediation analysis by path decomposition of the relationship between childhood adversity and psychopathology (internalising and externalising problems) at age 13 and persistent psychopathology. Candidate mediators were self-concept, parent-child relationship and hobby participation at age 9. Childhood adversity was reported by 28.2% of participants, and was significantly associated with internalising and externalising problems. Parent-child conflict mediated the relationship between childhood adversity and both age 13 and persistent psychopathology, accounting for 52.4% of the relationship between childhood adversity and persistent externalising problems (indirect odds ratio, 1.30; 95% CI 1.19-1.43) and 19.2% for persistent internalising problems (indirect odds ratio, 1.24; 95% CI 1.15-1.34). There was a small mediating effect of self-concept. Hobby participation and positive parent-child relationship did not mediate these relationships. Parent-child conflict explains almost half the relationship between childhood adversity and persisting externalising problems in adolescence, and a fifth of the relationship with persisting internalising problems. This suggests parent-child conflict is a good target for interventions in childhood to prevent adolescent psychopathology. None.

Mixed Mode I-II Fracture Path and Initiation Angle of Concrete at Mesoscale Level

This paper presents the mesolevel numerical simulation and experimental verification of mixed mode I-II fracture for the four-point shearing beam with a single-side notch based on the concrete damaged plasticity (CDP) method and the Monte Carlo simulation (MCS) random aggregate method. And it discusses the influence of volume rate and maximum size of the coarse aggregate on the fracture path and the initiation angle of concrete mixed mode I-II fracture. The results show that the coarse aggregate has a significant influence on the distribution width, average length, and the dispersion degree of the fracture path, the distribution of which can be fitted by a normal distribution function. Also, the distribution width of the fracture path has a quadratic function relationship with the height along the failure direction. The distribution of coarse aggregates at the top of the reserved notch also has a significant influence on the initiation angle, which determines the initiation crack direction of mixed mode I-II fracture in concrete. The initiation angle distributes near the failure direction and follows a normal distribution function.

Minimum Swept-Path Control for Autonomous Reversing of a Tractor Semi-Trailer

This paper presents a new approach for autonomous reversing of a tractor-semitrailer vehicle called minimum swept path control (MSPC). MSPC improves the performance of previous path following control (PFC) strategies, by minimizing the maximum excursion of the vehicle, while guaranteeing acceptable path following accuracy. A linear controller is devised, combining state feedback control with an optimized preview distance. A relationship between the maximum lateral offsets of the front axle of the tractor unit and the rear axle of the last trailer and the corresponding weights in the control cost function is established, and the controller is tuned to provide the best performance compromise. Several simulations of a tractor-semitrailer case are run, and the results show that the overall swept path width is reduced by more than 40% when compared with PFC methods.

Analysis of Bicycle Headway Distribution, Saturation Flow and Capacity at a Signalized Intersection using Empirical Trajectory Data

Predicting the bicycle flow capacity at signalized intersections of various characteristics is crucial for urban infrastructure design and traffic management. However, it is also a difficult task because of the large heterogeneity in cycling behavior and several limitations of traditional capacity estimation methods. This paper proposes several methodological improvements, illustrates them using high-resolution trajectory data collected at a busy signalized intersection in the Netherlands, and investigates the influence of key variables of capacity estimation. More specifically, it shows that the (virtual) sublane width has a significant effect on the shape of the headway distribution at the stop line. Furthermore, a new method is proposed to calculate the saturation headway (a key variable determining capacity), which excludes the cyclists initially located close to the stop line using a distance-based rule instead of a fixed number (as is usually done in practice). It is also shown that the saturation headway is quite sensitive to the sublane width. Moreover, a new, empirically based method is proposed to identify the number of sublanes that can be accommodated in a given cycle path, which is another key influencing variable. This method yields considerably lower estimates of the number of sublanes than traditional methods, which rely solely on the (available) cycle path width. Finally, the authors show that methodological choices such as the sublane width and the method used to estimate the number of sublanes have a considerable effect on capacity estimates. Therefore, this paper highlights the need to define a sound methodology to estimate bicycle flow capacity at signalized intersections and proposes some steps to move toward that direction.

Capacity, Capacity Drop, and Relation of Capacity to the Path Width in Bicycle Traffic

Bicycle usage is encouraged in many cities because of its health and environmental benefits. As a result, bicycle traffic increases which leads to questions on the requirements of bicycle infrastructure. Design guidelines are available but the scientific substantiation is limited. This research contributes to understanding bicycle traffic flow by studying the aggregated movements of cyclists before and after the onset of congestion within the setting of a controlled bottleneck flow experiment. The paper quantitatively describes the relation between capacity and path width, provides a qualitative explanation of this relation by analyzing the cyclist configuration for different path widths, and studies the existence of a capacity drop in bicycle flow. Using slanted cumulative curves and regression analysis, the capacity of a bicycle path is found to increase linearly with increasing path width. A steady drop in flow rate is observed after the onset of congestion, indicating that the capacity drop phenomenon is observed in bicycle traffic. The results presented in this paper can help city planners to create bicycle infrastructure that can handle high cyclist demand.

On the Pendant Number of Some New Graph Classes

Abstract A decomposition of a graph is a collection of its edge disjoint sub-graphs such that their union is . If all the sub-graphs in the decomposition are paths, then it is a path decomposition. In this paper, we discuss the pendant number, the minimum number of end vertices of paths in a path decomposition of a graph. We also determine this parameter for some graph classes. Keywords: Decomposition, path decomposition, pendant number . MSC2010: 05C70, 05C38, 05C40 Cite this Article Jomon K. Sebastian, Joseph Varghese Kureethara, Sudev Naduvath, Charles Dominic. On the Pendant Number of Some New Graph Classes. Research & Reviews: Discrete Mathematical Structures . 2019; 6(1): 15–21p.

Finding small-width connected path decompositions in polynomial time

A connected path decomposition of a simple graph G is a path decomposition (X1,…,Xl) such that the subgraph of G induced by X1∪⋯∪Xi is connected for each i∈{1,…,l}. The connected pathwidth of G is then the minimum width over all connected path decompositions of G. We prove that for each fixed k, the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game.

Stochastic Track Model for Tornado Risk Assessment in the U.S.

This paper presents the development of a stochastic tornado simulation model for the United States (US). The continental of the US is subjected to more than 1,000 tornadoes each year, causing significant financial losses and social disruption. Compared to hurricanes, the damage region of a tornado is relatively small and the probability of occurrence at a given location is extremely low. Therefore, it is not feasible to use solely the observed data or tracks to quantify the tornado risk for a given structure or a city that has not been affected by historical tornadoes. In this paper, a methodology for performing stochastic simulation of tornado tracks for the US is presented. The stochastic simulation framework consists of a genesis model, which utilizes the kernel density estimation to simulate the spawn locations of tornadoes. Statistical models for tornado parameters such as track length, path width and intensity, were calibrated using the tornado database maintained by the US National Oceanic and Atmospheric Administration (NOAA) Storm Prediction Center (SPC). The developed statistical models were used to simulate 1,000,000 years of tornado tracks. The simulated tornado parameters include the tornado occurrence rate, intensity (EF-scale), location, touchdown time, path length and path width. All these parameters are geographic dependent, meaning the parameters vary depending on the tornado spawn locations. The simulated spawn rates and other key parameters for the continental of the US are compared to the observations. Good agreements are observed between simulations and observations. To illustrate a potential use of the simulated tornado track database, a probabilistic tornado hazard analysis was performed for Moore, Oklahoma. The 50-year tornado hazard curves for three domain sizes are developed to assess the influence of the domain size on tornado risk.

Lane width‐based cellular automata model for mixed bicycle traffic flow

AbstractIn this paper, an improved extended Burgers cellular automata (EBCA) model for heterogeneous bicycle traffic flow is proposed. The compressibility of bicycle traffic flow is considered as a probability, which is related to the bicycle path width. The proposed improved EBCA model is calibrated and validated using the field data collected from three sites in Hangzhou, China. The proposed model is compared with the original EBCA model that does not consider compressibility. The comparison results show that our improved model can simulate heterogeneous bicycle traffic flow better. Moreover, the proposed model overcomes the shortcoming of the other BCA models that only specific path widths can be simulated directly when lane width is determined. Lastly, the EBCA model performance in the congested regions is improved significantly.

Investigating factors that influence pedestrian and cyclist violations on shared use path: An observational study on the Brooklyn bridge promenade

Shared use paths such as the Brooklyn Bridge promenade are painted with the centerline marking to avoid conflicts between pedestrians and cyclists and provide safe and comfortable accommodation of both modes of travel. However, pedestrians and cyclists do not always comply with the regulation indicated by line markings and corresponding signs, and violations happen frequently on such shared use paths. This study aims to identify factors that influence the violation behavior on the Brooklyn Bridge promenade by using random intercept/coefficient models. Video data of 6 hours from three sites of the promenade were analyzed, and a total of 497 pedestrians and 397 cyclists who were close to the centerline marking were defined as observations. Factors such as individual characteristics, environment conditions, and road conditions were considered. The modeling results show pedestrian violations are more likely when pedestrians are heading to Brooklyn, doing exercise, or there are pedestrians of the same direction, cyclists of the same or opposite directions, and vendor booth or garbage bin in front of them. Cyclist violations are more likely when cyclists are heading to Brooklyn, the cyclist path is too narrow, or there are pedestrians of the same or opposite direction in front of cyclists. The space-related variables (i.e., the presence of vendor booth and garbage bin, location effects) highlight the importance of adequate path widths, and the significant influence of direction variable implies that making the right-hand rule clear for pedestrians would decrease the likelihood of violation occurrence.

Equivalence between pathbreadth and strong pathbreadth

We say that a given graph G=(V,E) has pathbreadth at most ρ, denoted pb(G)≤ρ, if there exists a Robertson and Seymour’s path decomposition where every bag is contained in the ρ-neighbourhood of some vertex. Similarly, we say that G has strong pathbreadth at most ρ, denoted spb(G)≤ρ, if there exists a Robertson and Seymour’s path decomposition where every bag is the complete ρ-neighbourhood of some vertex. It is straightforward that pb(G)≤spb(G) for any graph G. Inspired from a close conjecture in Leitert and Dragan, (2016), we prove in this note that spb(G)≤4⋅pb(G).

A novel structure design and fabrication method for low liquid consumption and high precision device of colorimeter in water quality detection

Colorimeters based on the Lambert Beer law is now being widely used in water quality detection system. In this paper, we reported a novel colorimeter with low liquid consumption and high precision by optimizing the design of its transition zone and detection area, which would improve the absorbance for specific elements. Combined with stereoscopic fabrication method, six parts including the entrance and exit parts of liquid, the fluid transition path, the detection area, the light intensity enhanced board and the accessories can be assembled on a miniaturized chip. Experimental results show that when the width of the main fluid path for the as-made system is 600 μm, the liquid consumption for every determination is 0.001 mg reducing agent, 0.003 mg chromogenic agent and 0.007 mL standard testing liquid, which was four times lower than traditional structures. The high precision and low liquid consumption have shown our colorimeter possessing the potential of improved absorbance and portable integrated structure for in situ water detection system.

Gallai’s path decomposition conjecture for triangle-free planar graphs

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌊(n+1)∕2⌋. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai’s Conjecture for graphs with maximum degree at most 5, and Botler et al. (2017) verified it for graphs with treewidth at most 3. In this paper, we verify Gallai’s Conjecture for triangle-free planar graphs.

Investigation of inkjet printed path resistance in the context of manufacture and flexible application

Purpose The purpose of this paper is to study the repeatability of path manufacturing in the drop on demand inkjet printing process and the influences of environmental and application factors on path resistance. Design/methodology/approach Paths were printed as multiline paths in packets one-, two- and three-layer paths on polyimide substrates using nanoparticle silver ink. The sintering conditions were determined experimentally. The paths were subjected to climatic and shock exposures and to bending processes. The resistance, profile and width of the paths were measured and analyzed. The temperature distribution for electrically heated paths was measured to identify the defects. Findings This research shows the repeatability of printing processes and identifies the sources that cause diversification in path parameters after the whole technological process. The influence of shock, climatic and mechanical exposures on path electrical properties is indicated. An effective method for identifying defects thermally is shown. Research limitations/implications The research could have limited universality by arbitrarily use of substrate material, ink, printhead, process parameters and kind of sample exposures. Practical implications The research includes practically useful information about the width, thickness, defects and resistances and their changes during a typical application for a path printed with different technological parameters. Originality/value This research presents the results of original empirical research on problems concerning the manufacture of paths with uniform parameters and shows how path parameters will change under exposures that may occur in a typical application. The research combines both production and application aspects.