\r\n\tWith this goal in mind, together with the US Prof. John M. Ballato and the InechOpen publishing house since 2011 we have published in 2011, 2013, 2015 and 2017 4 books of our serial “Optoelectronics” and the book “Excitons”, edited in 2018 by Prof. Sergei L. Pyshkin. Publishing the new book “Luminescence” we are pleased to note the growing number of countries participating in this undertaking as well as for a long time fruitfully cooperating scientists from the United States and the Republic of Moldova. \r\n\tSpecialists from all over the world have published in edited by us books their works in the field of research of the luminescent properties of various materials suitable for use in optoelectronic devices, the development of new structures and the results of their application in practice.
",isbn:null,printIsbn:null,pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"5c69633b41645990265c48b109a7a23f",bookSignature:"Prof. Sergei L. Pyshkin",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/7341.jpg",keywords:"Semiconductors, Light-Emissive Device, Spectrometers, Boxcar-Integrators, Characteristics of Luminescence, Kinetics of Luminescece, Multiquantum Light Absorption, Superluminescence, Stimulated Emission, Light-Emissive Diods, Lasers, Methods and Devices, Transfer of Luminescence",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"April 24th 2018",dateEndSecondStepPublish:"May 15th 2018",dateEndThirdStepPublish:"July 14th 2018",dateEndFourthStepPublish:"October 2nd 2018",dateEndFifthStepPublish:"December 1st 2018",remainingDaysToSecondStep:"3 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"43016",title:"Prof.",name:"Sergei",middleName:"L.",surname:"Pyshkin",slug:"sergei-pyshkin",fullName:"Sergei Pyshkin",profilePictureURL:"https://mts.intechopen.com/storage/users/43016/images/system/43016.jpeg",biography:"Dr.Sc. Prof. Sergei L. Pyshkin is Scientific Advisor of the Institute of Applied Physics, Academy of Sciences of Moldova, Adjunct Professor and Senior Fellow of Clemson University, SC, USA, member of The US Minerals, Metals & Materials Society (TMS), awarded the State Prize of Republic of Moldova for investigations in solid-state physics and microelectronics.\r\n\r\nThe works deal with non-linear optics (multi -quantum absorption), electron and phonon transport phenomena, photoconductivity and light scattering, luminescence, crystal and thin film growth, molecular beam and laser assisted epitaxies, nanotechnology, lasers for medicine, scientific instrument making (boxcar-integrators and solid-state IR matrix photoreceivers). \r\n\r\nBiography of Prof. Pyshkin is included into the Marquis 'Who’s Who in America” (2008-2013) and 'Who’s Who in the World” (2009-Present).",institutionString:"Institute of Applied Physics, Academy of Sciences of Moldova",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"5",totalChapterViews:"0",totalEditedBooks:"5",institution:{name:"Academy of Sciences of Moldova",institutionURL:null,country:{name:"Moldova"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"14",title:"Materials Science",slug:"materials-science"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"247865",firstName:"Jasna",lastName:"Bozic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/247865/images/7225_n.jpg",email:"jasna.b@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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1. Introduction
The demand for off-season cultivation of fruits and vegetables require different aspects of automation and robotics in closed-field plant production environments like greenhouses [1]. Modern greenhouse bioproduction systems are required to exhibit integration of automation, biological culture practices, and control systems through the concept of Automation-Culture-Environment-oriented SYStems analysis (ACESYS) as defined in [2, 3]. The growth condition for Solanaceae vegetables in the greenhouse provides the leeway for the growth of other plants as well. In greenhouse cultivation of Cucumber (Cucumis sativus), the growth of weeds like cleavers, amaranth, camelthorn, grass quack, and oat wild decreases the final crop yield and quality. These weeds compete with cucumbers for nutrients, water, and photosynthesis. During the growing period, weeds use a large portion of water and nutrient, and because of their physiological properties, they grow simultaneously and rapidly with the original plant. It is, therefore, necessary to eliminate them before causing serious damage to the original plants. Various mechanical and chemical methods, as well as cultivation techniques, have been proposed to prevent the growth of weeds, including mechanical techniques, hand picking, spraying, environment heating, herbicides and biocontrols, and soilless cultural practices. For example, weed biocontrol is the suppression of weeds by insects and microorganisms that feed on the target plants or otherwise parasitize them. The success in this method is not always guaranteed because biocontrol is species specific, and there are hundreds of serious weed species. Cultural control includes those management practices that modify the agro-ecosystem to make the pasture, crop, or forest ecosystem resistant to weed establishment, i.e., integrating sheep or goats to browse brush species and fowl to graze herbs and grasses [4]. Prior to the development of modern herbicides, rancher and forest managers relied mainly on mechanical methods of weed control, such as grubbing, bulldozing, dragging, cabling, and mowing. Compared to mechanical weed control methods, herbicides are more effective at a lower cost. Herbicidal weed control results in greater grass production in pastures than does clipping of weeds [5]. In order to apply chemical directly to the weed’s vascular tissue, a direct chemical application end effector is required to cut the weed’s stem and spread the chemical on the cut surface. An example of such application can be found in [6] where a prototype weed control robot was developed to spray weeds in cotton plants in the seed line. A real-time intelligent weed control system was introduced in [7] for selective herbicide application to in-row weeds using machine vision and chemical application. A minirobot to perform spraying activities based on machine vision and fuzzy logic has been described in [8, 9]. More examples of autonomous vehicle robot for spraying the weeds can be found in [10, 11, 12].
The use of labor force that manually pulls out the weeds is still practiced by local growers. This is, however, not an efficient method since the availability of the skilled workforce that accepts repetitive tasks in the harsh greenhouse and field conditions impose uncertainties and timeliness costs [13]. It is, therefore, necessary to select a proper method for effective weed control. The trends in the agricultural robotics in the past 10 years show that automation of plant trimming with simultaneous localization and mapping techniques will change the industry in future [14]. The available time, labor, equipment, costs, and types of weeds and the areas infested need to be considered when planning a weed control program. In this regard, agricultural robotic and automation technology plays an essential role in improving the interactions between human, machine, and plants [15]. For example, the prevention of musculoskeletal disorders in manual harvesting operations in Dutch greenhouses has motivated various researchers for replacement of human labor by automatons robot for picking cucumber [16] and sweet pepper [13] fruits. Automation is a viable and sometimes necessary method to ensure maximum profits with minimum costs. In fact, one of the main purposes of agricultural automation has been always concerned with the substitution of human workforce by robots or mechanized systems that can handle the tasks more accurately and uniformly at a lower cost and higher efficiency [17, 18, 19, 20, 21, 22].
Research and development in agricultural robotics date back to 1980s, with Japan, the Netherlands, and the USA as the pioneer countries. Example of such research works included the works of [7, 23] for robotic weed control and automated harvesting of tomato. Development of an autonomous weeding machine requires a vision system capable of detecting and locating the position of the crop. Such vision system should be able to recognize the accurate position of the plant stem and protects it during the weed control [24]. A near-ground image capturing and processing technique to detect broad-leaved weeds in cereal crops under actual field conditions has been reported in the work of [25]. Here, the researchers proposed a method that uses color information to discriminate between vegetation and background, while shape analysis techniques were applied to distinguish between crop and weeds. Shape features of the radish plant and weed were investigated by [26]. They proposed a machine vision system using a charge coupled device camera for the weed detection in a radish farm resulting 92% success rate of recognition for radish and 98% for weeds.
A combined method of color and shape features for sugar beet weed segmentation was proposed by [27] with 92% success rate in classification. This rate increased to 96% by adding two shape features. Another approach extracted a correlation between the three main color components R, G and B, which constitute weeds and sugar beet color classes by means of discriminant analysis [28]. Their method resulted in different classification success rates between 77 and 98%. The segmentation of weeds and soybean seedlings by CCD images in the field was studied by [29]. Texture features of weed species have been applied for distinguishing weed species by [30] with grass and broadleaf classification accuracies of 93 and 85%, respectively. Textural image analysis was used to detect weeds in the grass [31]. Gabor wavelet features of NIR images of apples were extracted for quality inspection and used as input to kernel PCA [32]. Kernel PCA first maps the nonlinear features to linear space, and then, PCA is applied to separate the image Gabor wavelet (5 scales and 8 orientations) combined with kernel PCA that had the highest recognition rate (90.5%). Improvements in vision-based control system [13, 33, 34, 35, 36] have enabled several applications of robotic manipulators for greenhouse and orchard tasks and have contributed to the decrease in workload and labor’s fatigue, while improving the efficiency and safety of the operations. These achievements were considered a challenge in the earlier agricultural robotics works [23, 37, 38]. For example, spray equipment for weed control has been developed with vertical spray booms that increase the deposition in the canopy [39, 40, 41]. Some of these alternatives are self-propelled vehicles such as Fumimatic® (IDM S.L, Almería, Spain) and Tizona (Carretillas Amate S.L., Almería, Spain), or autonomous vehicles such as Fitorobot (Universidad de Almería, Cadia S.L., Almería, Spain) that have been designed specifically to move without difficulty over loose soils and in spaces with a large number of obstacles [41]. These vehicles rely on inductive sensors to follow metal pipes buried in the soil. Few studies have addressed the navigation problem of vehicles in greenhouses operating completely autonomously [9, 11, 15]. The main challenge of these systems is that localization approaches needed for feeding the closed-loop controllers would lead to inaccurate measurements after a few steps fail for long trajectories [42]. A stereovision system along with an image processing algorithm was used to recognize the weeds and also to estimate their location in the field. In order to experiment with vision sensors and agricultural robots, [13] created a completely simulated environment in V-REP, ROS, and MATLAB for improvement of plant/fruit scanning and visual servoing task through an easy testing and debugging of control algorithms with zero damage risk to the real robot and to the actual equipment. In another study, [43] designed a field survey mobile robot platform based for navigating inside greenhouses and open-field cultivation for automated image acquisition. A functional model shown in Figure 1 was introduced by [44] in the field test of an autonomous robot for deleafing cucumber plants grown in a high-wire cultivation system. This model was also adapted and used by [13] for the robotic harvesting of sweet pepper and on a greenhouse field survey mobile platform [43]. Artificial neural networks have also been used by many researchers to discriminate weeds [45, 46] with machine vision as shown in Figure 2. A fixed-position weed robot was presented by [47], which is interfaced to a standard belt-conveyor displacement system and provides the robot with pallets containing the crops. These reviews indicate that a commercial robotic platform for the elimination of weeds in a cucumber greenhouse has not been materialized yet. In addition, most of the research works in the area of robotic weed control are applicable prior to the plant growth or in some cases when the main plant height is between 0.2 and 0.3 m.
Figure 1.
Task sequence during leaf picking of cucumber, adapted from [44].
Figure 2.
Mechatronic paradigm followed in this research, adapted from [41].
The overall objective of this study was to design and develop an affordable robotic weed control system for application in greenhouse cultivation of cucumbers where plants can reach to a height of 10 m. Our design is based on mechanical weed removal techniques without using chemical materials. The specific objectives were to determine (i) the best blade design for cutting the weeds among cultivation rows, (ii) the best blade rotation (BR) speed, and (iii) the best arm motor (AM) speed.
2. Materials and methods
2.1. Overview of the prototype robot weed
A flowchart of the methodology is shown in Figure 3. A prototype robot was designed using AutoCAD software 2011 v18.1 (Autodesk Inc., San Rafael, CA, USA). Schematic views of the prototype robot, as well as the corresponding dimensions and parts are shown and illustrated in Figures 4 and 5. The main mechanical components of the robotic platform consist of a monorail, main chassis, ball bearings, wheels, arms, blade, and adjusting mechanism. Major electrical components include DC motors, microswitches, a 12 V 7.2–9 amp sealed lead acid battery, SRF05 ultrasonic sensors, pic 18F4550 microcontroller, and 2 × 24 LCD monitor (Figure 6). We began with the design of a monorail that was responsible to support the robot navigations and stops between two cucumber rows inside the greenhouse. The monorail has a width of 0.06 m and was placed 0.4 m above the ground (Figure 5A and B). The algorithm for robot navigation between two consecutive stops points on the monorail is also illustrated in the flowchart of Figure 5. Right after the robot is switched on, it starts moving on the monorail that is fixed along the greenhouse from one row to another. Upon reaching the first stopper point on the rail, the robot strikes the first microswitch, which sends a deactivation signal to the first motor responsible for moving the robot. While stopped between two cucumber plant rows, the robot scans for weeds and determines the distance between the detected weed and the blade arm using the ultrasonic sensors. Subsequently, a command signal is sent to the arm motor and blade motor for activating the blade rotation as illustrated in the flowchart of Figure 5.
Figure 3.
Flowchart of the research methodology.
Figure 4.
The CAD model design of the weed control robot.
Figure 5.
The mechanism and flowchart for the robot navigation and control on the monorail showing (A) the monorail layout, (B) the robot mounted on the monorail, (C) the robot main body and manipulator arm, and (D) the joint setup between the robot and the monorail.
Figure 6.
Major electrical module and wiring connections of the weed control robot.
2.2. Design of the mechanical parts
The moving mechanical arm consists of a chassis, a small arm, and the main arm. Two main criteria were considered in designing the robot frame including minimum weight (for increasing the motor efficiency), and strength (for standing vibrations). The frame was made from an iron band bearing with the dimensions of 0.02 × 0.18 × 0.005 m. In order to provide support for the battery, bearing bases, microswitches, and the main arm, we installed additional extensions to the frame in a way that the robot gravity center is placed on the monorail. The battery is the heaviest part of the robot and can power the robot for 2 h. It was installed on the central frame above the rails and wheels. The battery weight creates stability for the robot when the main arm is outstretched, and this weight and location for the battery can hold the spinning wheel implemented in place. We placed several holes on the frame to facilitate the installation of the motor, wheels, and the required electrical fragments (Figure 5C and D). The robot makes use of four ball bearings of diameter 0.02 m, out of which three were used to hold the robot to the rail and to facilitate a smooth movement (two bearings were placed on the right and one on the left side). The fourth bearing was used to act as the second wheel for the robot. All the ball bearings have a diameter of 0.02 m and are installed on the central frame. The diameter of the robot main wheel is 0.04 m, and the ideal speed was determined using trial and errors and time-motion studies during the conducted tests. The arm frame is made of an iron band bearing with a dimension of 0.02 × 0.2 × 0.005 m. A blade was installed on the main arm that moves forward and enables robot access to the weeds between the main plants. A shank protector in one of the holes in the arm frame makes the movement and the selection of the angle for smooth cutting.
2.3. Design of the electronic parts: sense and action mechanism
Major electronic components of the robot are three sets of SRF05 ultrasonic sensors, a PIC18F4550 microcontroller, and a 2 × 24 LCD monitor (Figure 6). The ultrasonic sensors were placed in a row having 0.10 m distance from each other. The sensors are specially positioned in a way that they cover the space between two cucumber plants on the cultivation row. As mentioned earlier, upon receiving a signal indicating weed existence, the microcontroller program determines the distance between the weed and the sensors and whether the weed is on the left, right, or middle of the sensors. This signal activates the cutting mechanism. Finally, the information of the entire process, including the distance between weed and sensors, and the specific sensor that identified the weed are shown on the robot LCD. During the experimental phase, we considered several improvements and adjustment on the sensing part and corresponding microcontroller program. For example, we used a tube pipe cover for each of the ultrasonic sensors to change the circular waves to linear waves. This was necessary because sound waves that broadcast from transmitters of ultrasonic sensors are circular. When these sensors are close to the ground, the broadcasting waves that bounce off from the ground are misinterpreted as weeds.
The robot movements are supported by three 12 V, 0.89 A DC motors that are labeled for this paper by motor 1, 2, and 3. The first motor was fixed directly to the wheels in front of the robot and was responsible for the robot movement on the monorail. To select the optimum speed for the robot, six motor speeds of 30, 40, 50, 60, 80, and 120 rpm were tested. We found that the motor with 60 rpm, 1.358 N·m torque, 12 V, 0.89 A had the best performance in the greenhouse under study. The second motor was connected to the small arm and is responsible to rotate the big arm that moves the blade of the robot at a selected speed of 10 rpm and torque of 8.15 N·m. The third motor was fixed to the frame of the main arm for rotating the blade at a high speed of 3500 rpm for efficient weed cutting and removal. This frame can move up and down and can fix the distance between the blade and the ground level. It should be noted that the 3500 rpm blade rotation speed and the 10 rpm arm motor speed were found from the experiments.
2.4. Blade design and analysis
Three types of blade, namely the S-shaped, the triangle-shaped, and the circular-shaped blade (Figure 7) were initially considered in the weed cutting experiments. We conducted several tests to find the best blade width (equal to 0.1 m) for matching the 0.4 m distance between two cucumber plants. Based on our field tests, we found that the S-shaped blade was the most efficient design for the purpose of weed cutting. The blade was built from double stainless steel material to resist the corrosion in high humidity greenhouse environment. Analysis and calculations were carried out for finding the blade tip speed and corresponding vector components according to the formulations given in [48]. The corresponding diagrams of this analysis are shown schematically in Figure 7. It can be observed from Figure 7A that the direction of the tip of the blade follows a cycloid curve on the ground level. The component of blade speed in the direction of robot forward speed vector, as well as the demonstration of vector gradient in the blade speeds, is shown in Figure 7B–D. Here, WB is the circular speed of the blade (rad/s), Vf is the forward speed of robot [m/s], Vbf = Vf + Vb is the ratio of the total speed of blade to ground [m/s], vb = rb × WB is the circumferential speed of blade [m/s], rb is the radius of blade [m], U represents the direction of the robot movement, and V is the linear speed of blade [m/s]. The speed of the tip of the blade on the ground is equal to the sum of robot forward speed and its circumferential speed. Having the direction of robot moving (U), the direction of the moving blade will be in the direction of Vbf, which changes its direction as the blade rotates in the time frame t [s]. Therefore, to find the components of U and V, the speed of the blade tip can be written as the component of blade speed in direction of moving U [48], that is Vu = Vf – rb × WB Sin (θ), and Vv = Vbv = Vb × Cosθ = rb × WBcos (θ), where θ = WB × t is the angle between blade and movement direction, Vu and Vu are the speed component [m/s], and t [s] is the measured time from the initial angle θ = 0. Therefore, the speed of the blade tip with respect to the ground is calculated as Vbf=Vv2+Vu2. Figure 7E shows forces and torque vectors of the cutting strike on the weed stem. Here, the force fb [N] is the bending strength of the plant body, fr is the cutting force [N], Ip is the pant geometry hardness torques, and mp is the weight of the cutting part of the plant [kg]. During the trial and error experiments, it was found that a minimum strike speed of between 50 and 75 m/s is required for cutting the weeds.
Figure 7.
Design of the cutting blade, (A): calculating the velocity of rotating blade in stickles, (B): component of blade speed in direction of moving U, (C, D): calculating U and V, (E): components of forces, and (F): vector demonstration of the blade speed. Adapted from [48].
2.5. Experiment setup
The weed control robot was tested in a 5000 m2 greenhouse in Jiroft city (28°40′41″N 57°44′26″E) located to the south of Kerman province of Iran (Figure 8). We planted over 10,000 cucumber seeds in pots and placed them in the greenhouse with spaces between the two plants being 0.4 m. It should be noted that in order to manually remove the weeds from 1 ha of the greenhouse under study, four seasonal workers had to perform the task every day, for 8 months (equivalent to 832 man/hour). Three experiments were conducted at different growth stages as follows: (i) during the seedling and germination stage, 15 days after the crop was cultivated and the surrounding weeds were also 15 days old (these weeds usually have thin and very flexible stalks and are 10 cm high), (ii) during the vegetation and early fruiting stage, when the cucumber plants were 2 months old, and (iii) during the mature fruiting stage, when the plants were at their mature height. Three types of blades were selected, namely the S-shaped, triangular-shaped, and circular-shaped blade. For each blade, we assigned three blade rotation (BR) speeds of BR1 = 3500, BR2 = 2500, and BR3 = 1500 rpm with two arm motor (AM) speed of AM1 = 10 and AM2 = 30 rpm. A factorial design with two-way analysis of variance (ANOVA) was used to determine variation effects in the cutting weed performance of each blade due to BR speed, AM speed, and their interaction. For the kth blade type, under the ith level of blade speed and the jth level of arm speed factor, the two-way ANOVA model was stated as Yijk=μ+bi+aj+b.aij+εijk, where Yijk is the dependent variable representing the percentage of weeds cut (PWC) in an experiment. Time and motion study was conducted for the robot to move from one stopper to another. For motor no. 1, with a typical rotational speed of 60 rpm, and the wheel diameter of 0.04 m, the forward speed of the robot (VF) becomes 0.1256 m/s. Hence, the required time T [s] for the robot to travel the distance of X = 0.40 [m] between two consecutive stoppers is equal to T = 3.2 s using Eq. (1). The possibility for the robot to pass through the two stoppers within a row was considered for the consequent calculations. For the arm motor, the typical speed is 10 rpm, which implies that it takes T = 6 s for the robot to remove the weed between two plants.
T=3.6×XVFE1
Figure 8.
Outside and inside views of the experimental site (top), and corresponding factorial design of experiment (bottom) for determining the best combination of blade type, blade speed, and arm speed.
3. Results
Results of statistical analysis are summarized in Tables 1–3 showing that the effects of blade type (T), blade rotation (BR) speed, and arm motor (AM) speed are significant at the 0.05 level. Moreover, it was found that the S-shaped blade with a mean (μ) of 67.8% and standard error (σ) of 3.052% had the highest effect, and triangular-shaped blade with μ = 61.38% and σ = 3.083% had the lowest effect on the percentage of the weeds cut (PWC). The BR factor was significant at P < 0.05, indicating that blade rotation speed of 3500 rpm with μ = 78.23% and σ = 1.71% had the highest effect and the 1500 rpm with μ = 50.39 and σ = 1.86% had the lowest effect. The AM speed factor was also found to be significant at P < 0.05, which indicates that the speed of 10 rpm with μ = 69.1% and σ = 2.45% had the highest effect and the speed of 30 rpm with μ = 59.8% and σ = 2.46% has had the lowest effect on the PWC. It was found that (Table 1) different blade shapes with the AM speed of 10 rpm had a significant effect on the PWC. While the mean PWC by the S-shaped blades was the highest, increasing AM speed to 30 rpm reduced the efficiency of the S-shaped blade (as well as with the other two blades), resulting a mean PWC of 59.39%. According to the P-values in Table 2, while all of the main effects of blade type, BR, and AM speeds are significant at 0.05 level, their interactions were not found to have a significant effect on the PWC. The results provided in Table 3 show that the difference between the two blades, S-shape and triangular shape is significant at the 0.05 level. In other words, the mean of weeds cut by these two blades are significantly different, and according to the mean differences column, the mean of the PWC by the S-shape blade is larger than the PWC by the triangular-shaped blade. The mean difference between the S-shaped and the circular-shaped blade with P-value of 0.036 is also significant at the 0.05 level. This implies that the average PWC by these two blades are significantly different, and according to the mean differences column, the mean PWC by the S-shaped blade is larger than the mean of the PWC by the circular-shaped blade. It was found that the difference between the means of the triangular-shaped blade and circular-shaped blade with the P-value of 0.076 is not significant at the 0.05 level, that is, the mean of the PWC by these two blade types are not significantly different.
Blade type
μ: Mean percentage of weeds cut (%)
σ: Std. error (%)
A: S shaped
67.8
3.05
B: Triangular shape
61.38
3.08
C: Circular shape
64.3
3.38
Blade rotation speed (rpm)
1500
50.3
1.86
2500
64.9
1.51
3500
78.2
1.71
Arm motor speed (rpm)
10
69.148
2.457
30
59.88
2.461
Table 1.
Factor effects on the percentage of weeds cut.
Model
Sum of squares
Mean sum of squares
P-value
Blade type (T)
110.3
110.3
0.0462
Blade rotation speed (BR)
6977.1
3488.6
0.000
Arm motor speed (AM)
1157.4
1157.4
0.000
Error
1264.6
26.3
Interaction types
P-value
T × BR
114.1
57.1
0.1272
T × AM
5.4
5.4
0.6493
BR × AM
73.6
36.8
0.2560
T × BR × AM
16
8
0.7356
Table 2.
Variance analysis and effects of the robot blade type (T), blade rotation (BR) speed, and arm motor (AM) speed on the percentage of weed cutting performance.
Mean differences
P-value
Blade type
A-B
6.4444
0.000
A-C
3.5000
0.036
B-C
−2.9444
0.076
Blade rotation (rpm)
BR1-BR2
−14.5556
0.000
BR1-BR3
−27.8333
0.000
BR2-BR3
−13.2778
0.000
Table 3.
Comparison of significant difference between blade types (A: S-shape, B: Triangular shape, and C: Circular shape), and blades rotation speed (BR1: 1500, BR2: 2500, and BR3: 3500 rpm).
Results of analysis of variance also showed that the mean differences between the BR speeds are significant, indicating that the resulted PWC with BR1 = 1500, BR2 = 2500, and BR3 = 3500 rpm are not equal. More specifically, the PWC in 1500 rpm was found to be smaller than those of 2500 and 3500 rpm. In addition, the mean PWC in 2500 rpm was also smaller than that of 3500 rpm. This can also be observed from the bar plots of Figure 9, showing that the mean PWC in 1500 rpm is the smallest (59.39%) and that of 3500 rpm was the largest (78.23%). The bar plots in Figure 10 illustrate descriptive statistics and frequency of the PWC for the experiments with the robot using all factors (blade types A, B, C, blade rotation speeds of 1500, 2500, 3500 rpm, and arm motor speed of 10 and 30 rpm). It can be seen from Figure 10 that the average PWC by the blades was significantly different. Consequently, the highest PWC cut was related to S-shaped at the blade rotation speed of 3500 rpm. In each motor arm speed, the increase in the rotational blade speed caused an increase in the PWC. In each rotational blade speed, if the motor arm speed increases, the PWC cut will decrease. Comparing the interactions between the three different types of blades, blade speed, and the speed of the arm the following results was obtained: the highest PWC in the entire experiment was 95%, which was obtained when the S-shaped blade at the rotational speed of 3500 rpm was used and motor speed was 10 rpm. The lowest PWC was 45%, which was obtained when the blade speed was 1500 rpm, AM speed was 30 rpm, and the blade type was triangular in shape. The analysis of the interaction of the BR speed and blade type showed that (i) none of the mutual interactions was significant in the variance test, (ii) t-test showed that if the rotational speed of the blade is low, the blade type will have a significant effect on the PWC, and (iii) for all the blade types, the highest PWC cut was at BR speed of 3500 rpm.
Figure 9.
Comparison of the effects of various blade types on (left) and various blade rotation speeds (right) on the percentage of weeds cut.
Figure 10.
Bar plots describing percentage of weeds cut with different blade type, blade rotation speed, and robot arm speed.
4. Conclusion
In this study, we designed, developed, and fabricated a prototype robot for mechanical weed control in greenhouse cultivation of cucumber. Automatic weed cutting experiments that were carried using the robot consist of ultrasonic sensor, which senses the existence of weeds between the cucumber plants. The robot then moves between cucumber rows on a monorail in the greenhouse, with an arm that moves the blade between the plants for cutting the detected weeds. The entire process of weed detection, moving the arm and blades, and weeds cutting is carried out in 10 s. Among the three blade types tested (S-, triangular-, and circular shapes), it was concluded that the S-shape was the most efficient design. For the best blade rotation (BR) and arm motor (AM) speeds, it was concluded that as the AM speed increased, the percentage of weeds cut (PWC) reduces; therefore, the motor with 10 rpm, 8.15 N·m torque, 12 V, and 0.89 A was selected to for moving the arm. The average weeds cut at 10 and 30 rpm was 69.1 and 58.9%, respectively. Finally, it was concluded that the best robot performance corresponding to the highest percentage of weeds cut was achieved with the S-shaped blade when the BR speed was 3500 rpm, and the AM speed was 10 rpm.
Conflict of interest
The authors declare no conflict of interest.
\n',keywords:"agricultural robot, weed control, cucumber, greenhouse",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/63478.pdf",chapterXML:"https://mts.intechopen.com/source/xml/63478.xml",downloadPdfUrl:"/chapter/pdf-download/63478",previewPdfUrl:"/chapter/pdf-preview/63478",totalDownloads:694,totalViews:258,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"March 31st 2018",dateReviewed:"August 16th 2018",datePrePublished:"November 5th 2018",datePublished:"January 3rd 2019",dateFinished:null,readingETA:"0",abstract:"A prototype robot that moves on a monorail along the greenhouse for weed elimination between cucumber plants was designed and developed. The robot benefits from three arrays of ultrasonic sensors for weed detection and a PIC18 F4550-E/P microcontroller board for processing. The feedback from the sensors activates a robotic arm, which moves inside the rows of the cucumber plants for cutting the weeds using rotating blades. Several experiments were carried out inside a greenhouse to find the best combination of arm motor (AM) speed, blade rotation (BR) speed, and blade design. We assigned three BR speeds of 3500, 2500, and 1500 rpm, and two AM speed of 10 and 30 rpm to three blade designs of S-shape, triangular shape, and circular shape. Results indicated that different types of blades, different BR speed, and different AM speed had significant effects (P < 0.05) on the percentage of weeds cut (PWC); however, no significant interaction effects were observed. The comparison between the interaction effect of the factors (three blade designs, three BR speeds, and two AM speeds) showed that maximum mean PWC was equal to 78.2% with standard deviation of 3.9% and was achieved with the S-shape blade when the BR speed was 3500 rpm, and the AM speed was 10 rpm. Using this setting, the maximum PWC that the robot achieved in a random experiment was 95%. The lowest mean PWC was observed with the triangular-shaped blade (mean of 50.39% and SD = 1.86), which resulted from BR speed of 1500 rpm and AM speed of 30 rpm. This study can contribute to the commercialization of a reliable and affordable robot for automated weed control in greenhouse cultivation of cucumber.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/63478",risUrl:"/chapter/ris/63478",book:{slug:"agricultural-robots-fundamentals-and-applications"},signatures:"Amid Heravi, Desa Ahmad, Ibrahim A. Hameed, Redmond Ramin Shamshiri, Siva K. Balasundram and Muhammad Yamin",authors:[{id:"182449",title:"Prof.",name:"Ibrahim",middleName:null,surname:"A. Hameed",fullName:"Ibrahim A. Hameed",slug:"ibrahim-a.-hameed",email:"ibib@ntnu.no",position:null,institution:{name:"Norwegian University of Science and Technology",institutionURL:null,country:{name:"Norway"}}},{id:"203413",title:"Dr.",name:"Redmond Ramin",middleName:null,surname:"Shamshiri",fullName:"Redmond Ramin Shamshiri",slug:"redmond-ramin-shamshiri",email:"rshamshiri@atb-potsdam.de",position:null,institution:{name:"Leibniz Institute for Agricultural Engineering Potsdam-Bornim",institutionURL:null,country:{name:"Germany"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Materials and methods",level:"1"},{id:"sec_2_2",title:"2.1. Overview of the prototype robot weed",level:"2"},{id:"sec_3_2",title:"2.2. Design of the mechanical parts",level:"2"},{id:"sec_4_2",title:"2.3. Design of the electronic parts: sense and action mechanism",level:"2"},{id:"sec_5_2",title:"2.4. Blade design and analysis",level:"2"},{id:"sec_6_2",title:"2.5. Experiment setup",level:"2"},{id:"sec_8",title:"3. Results",level:"1"},{id:"sec_9",title:"4. Conclusion",level:"1"},{id:"sec_13",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Shamshiri RR, Kalantari F, Ting KC, Thorp KR, Hameed IA, Weltzien C, et al. Advances in greenhouse automation and controlled environment agriculture: A transition to plant factories and urban agriculture. International Journal of Agricultural and Biological Engineering. 2018;11(1):1-22'},{id:"B2",body:'Shamshiri RR, Mahadi MR, Thorp KR, Ismail WIW, Ahmad D, Man HC. Adaptive management framework for evaluating and adjusting microclimate parameters in tropical greenhouse crop production systems. In: Jurić S, editor. Plant Engineering. Rijeka: InTech; 2017'},{id:"B3",body:'Ting KC, Lin T, Davidson PC. Integrated urban controlled environment agriculture systems. In: Kozai T, Fujiwara K, Runkle ES, editors. LED Lighting for Urban Agriculture. 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St. Joseph, Michigan, USA: American Society of Agricultural Engineers; 1987'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Amid Heravi",address:null,affiliation:'
Department of Biological and Agricultural Engineering, Faculty of Engineering, Universiti Putra Malaysia, Malaysia
Department of Farm Machinery and Power, University of Agriculture Faisalabad, Pakistan
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1. Introduction
For terminology and results in graph theory, we refer to [1, 2, 3, 4, 5]. For further notions and concepts on graph classes, graph operations, graph products and derived graphs, refer to [3, 6, 7, 8, 9, 10]. Unless mentioned otherwise, all graphs mentioned in this chapter are simple, finite, connected and undirected.
1.1 Basics of graph labelling
Labelling of a graph G can broadly be considered as an assignment of labels or weights to the elements (vertices and edges) of G subject to certain pre-defined conditions. The research on graph labelling has flourished in the second half of twentieth century after the introduction of the notion of β-valuations of graphs in [11]. The β-valuation of a graph G is an injective map f:VG→1,2,3…E such that the induced function f∗:EG→1,2,3…E, defined by f∗uv=∣fu−fv∣ for all uv∈EG, is also injective. Later, β-valuation of graphs was popularly known to be the graceful labelling of graphs (see [12]). Many variations of number valuations have been defined in the literature since then and most of those studies were based on the number theory and/or number theoretic properties of sets. For concepts and results in number theory, see [13, 14, 15, 16].
Analogous to the number valuations of graphs, the notion of set-labelling of graphs has been introduced in [17] as follows: Given a non-empty ground set X, a set-labelling or a set-valuation of a graph G is an injective function f:VG→PX, the power set of X, such that the induced function f⊕:VG→PX defined by f⊕uv=fu⊕fv, for all uv∈EG, where ⨁ is the symmetric difference of two sets. A graph which admits a set-labelling is called a set-labelled graph or a set-valued graph. If the induced function f⊕ is also injective, then the set-labelling f is called a set-indexer.
Subsequent to this study, many intensive investigations on set-labelling of graphs and its different types have been taken place. An overview of such studies on set-labelling of graphs can be seen in [17, 18, 19, 20, 21, 22, 23]. The study on set-labelling has been extended by replacing the binary operation ⨁ by some other binary operations of sets. For example, two different types of set-labellings—called disjunctive set-labelling and conjunctive set-labelling—of graphs have been studied in [24]. These set-labellings are defined respectively in terms of the union and the intersection of two sets instead of the symmetric difference of two sets.
1.2 Sumsets and integer additive set-labelled graphs
Integer additive set-labelling or sumset labelling of graphs has been a new addition to the theory of set-labelling of graphs recently. The notion of sumsets of two sets is explained as follows: Let A and B be two sets of numbers. The sumset of A and B is denoted by A+B and is defined by A+B=a+b:a∈Ab∈B (see [25]). Remember that a sumset of two sets can be determined if and only if both of them are number sets. If either A or B is countably infinite, then their sumset A+B is also a countably infinite set and if any one of them is a null set, then the sumset is also a null set. If C is the sumset of two sets A and B, then both A and B are said to be the summands of C.
We note that A+0=A and hence A and 0 are called the trivial summands of the set A. Also, note that A+B need not be a subset or a super set of A and/or B. But, A⊂A+B if 0∈B. Furthermore, the sumset of two subsets of a set X need not be a subset of the ground set X. These observations are clear deviations from the other common binary operations of sets and thus the study of sumsets becomes more interesting. For the terms, concepts and results on sumsets, we refer to [25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
Note that if A and B are two non-empty finite sets of integers, then ∣A∣+∣B∣−1≤∣A+B∣≤∣A×B∣=∣A∣∣B∣ (see [25]). The exact cardinality of the sumset A+B always depends on the number as well as the pattern of elements in both the summands A and B. The counting procedure in this case is explained in [35] as follows: Two ordered pairs ab and cd in A×B is said to be compatible if a+b=c+d. If ab and cd are compatible, then it is written as ab∼cd. It can easily be verified that this relation is an equivalence relation. A compatibility class of an ordered pair ab in A×B with respect to the integer k=a+b is the subset of A×B defined by cd∈A×B:ab∼cd and is denoted by abk or Ck. The cardinality of a compatibility class in A×B lies between 1 and minAB. Note that the sum of coordinates of all elements in a compatibility class is the same and this sum will be an element of the sumset A+B. That is, the cardinality of the sumset of two sets is equal to the number of equivalence classes on the Cartesian product of the two sets generated by the compatibility relation defined on it.
Using the concepts of the sumsets of sets, the notion of integer additive set-labelling of graphs has been introduced in [36] as follows: Let X be a set of non-negative integers and P0X be the collection of the non-empty subsets of X. Then, an integer additive set-labelling or an integer additive set-valuation of a graph G is an injective map f:VG→PX such that the induced function f⊕:EG→PX is defined by f+uv=fu+fv, for all uv∈EG, where fu+fv is the sumset of the set-label the vertices u and v (see [36, 37]). A graph with an integer additive set-labelling is called an integer additive set-labelled graph. It can very easily be verified that every graph G admits an integer additive set-labelling, provided the ground set X is chosen judiciously.
Following to the above path-breaking study, the structural properties and characteristics of different types of integer additive set-labellings of graphs are studied intensively in accordance with the cardinality of the set-label, nature and pattern of elements in the set-label, nature of the collection of set-label, etc. Some interesting and significant studies in this area can be found in [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Later, the studies in this area have been extended by including the sets of integers (including negative integers also) for labelling the elements of a graph. Some extensive studies in this area, can be seen in [45, 46].
As a specialisation of the sumset labelling of graphs, the notion of modular sumset labelling of graphs and corresponding results are discussed in the following section.
2. Modular sumset labelling of graphs
2.1 Basics of modular sumsets
Recall that Zn denotes the set of integers modulo n, where n is a positive integer. The modular sumset of two subsets A and B of Zn is the set k:a∈Ab∈Ba+b≡kmodn. Unless mentioned otherwise, throughout this chapter, the notation A+B denotes the modular sumset of the sets A and B. Unlike the ordinary sumsets, the modular sumset A+B⊆Zn if and only if A,B⊆Zn. This fact will ease many restrictions imposed on the vertex set-label of a sumset graph G in order to ensure that the edge set-label are also subsets of the ground set.
If we assign the null set Ø to any vertex as the set-label, the set-label of every edge incident at that vertex will also be a null set. To avoid such an embarrassing situation, we do not consider the null set for labelling any vertex of graphs. Thus, the set of all non-empty subsets of a set X is denoted by P0X. That is, P0X=PX\\0.
2.2 Modular sumset graphs
In view of the facts stated above, the modular sumset labelling of a graph is defined as follows:
Definition 2.1. [47] A modular sumset labelling of a graph G is an injective function f:VG→P0Zn such that the induced function f+:EG→P0Zn is defined as f+uv=fu+fv, where fu+fv is the modular sumset of the set labels of the vertices u and v. A graph which admits a modular sumset labelling is called a modular sumset graph.
Definition 2.2. [47] A modular sumset labelling of a graph G is said to be a uniform modular sumset labelling of G if the set-label of all its edges have the same cardinality. A modular sumset labelling f of G is said to be a k-uniform modular sumset labelling if f+uv=k,∀uv∈EG.
Proposition 2.1. [47] Every graphGadmits modular sumset labelling (for a suitable choice ofn).
The proof of the above proposition is immediate from the fact that fu+fv⊆Zn if and only if fu,fv⊆Zn.
An immediate question that arises in this context is about the minimum size of the ground set Zn (that is, the minimum value of n) required for the existence of a modular sumset labelling of G.
As in the case of sumsets, the cardinality of the modular sumsets also attracted the attention. Hence, we have the bounds for the cardinality of an edge set-label of a modular sumset graph G is as follows:
Theorem 2.2. [47] Letf:VG→P0Znbe a modular sumset labelling of a given graphG. Then, for any edgeuv∈EG, we have
∣fu∣+∣fv∣−1≤∣f+uv∣=∣fu+fv∣≤∣fu∣∣fv∣≤n.E1
The theorem follows immediately from the theorem on the cardinality of sumsets (see Theorem 2.7, p. 52, [25]).
In this context, it is quite interesting to investigate whether the bounds are sharp. It has also been proved in [25] that the lower bound is sharp when both fu and fv are arithmetic progressions (we call set an arithmetic progression if its elements are in arithmetic progression) with the same common difference. We shall discuss the different types of modular sumset graphs based on the set-labelling numbers of its vertices and edges, one by one in the coming discussions.
3. Arithmetic modular sumset graphs
As mentioned above, the lower bound of the inequality (1) is sharp if both summand set-label are arithmetic progressions with the same common difference. If the context is clear, the common difference of the set-label (if exists) of an element may be called the common difference of that element. The deterministic ratio of an edge of G is the ratio, k≥1 between the common differences of its end vertices. In view of this terminology we have the following definition.
Definition 3.1. For any vertex v of G, if fv is an arithmetic progression, then the modular sumset labelling f:VG→P0Zn is called a vertex arithmetic modular sumset labelling of G. In a similar manner, for any edge e of G, if fe is an arithmetic progression, then the modular sumset labelling f:EG→P0Zn is called an edge arithmetic modular sumset labelling of G.
The difference set of a non-empty set A, denoted by DA, is the set defined by DA=a−b:ab∈A. Note that if A is an arithmetic progression, then its difference set DA is also an arithmetic progression and vice versa. Analogous to the corresponding result of the edge-arithmetic sumset labelling of graphs (see [44, 46]), the following result is a necessary and sufficient condition for a graph G to be edge-arithmetic modular sumset graph in terms of the difference sets the set-label of vertices of G.
Theorem 3.1.Letfbe a modular sumset labelling defined on a graphG. If the set-label of an edge ofGis an arithmetic progression if and only if the sumset of the difference sets of set-label of its end vertices is an arithmetic progression.
Proof. Let f:VG→Pℕ0 be a modular sumset labelling defined on G. Let ai,aj be two arbitrary elements in fu and let br,bs be two elements in fv. Then, ∣ai−aj∣∈Dfu and ∣ai−aj∣∈Dfu. That is, Dfu=ai−aj:aiaj∈fu and Dfv=br−bs:brbs∈fv.
Now, assume that f+e=f+uv is an arithmetic progression for an edge e=uv∈EG. That is, A=fu+fv is an arithmetic progression. Then, the difference set DA=a−b:ab∈A=fu+fv is also an arithmetic progression. Since a,b∈A, we have a=ai+br and b=aj+bs, where ai,aj∈fu and br,bs∈fv. Then,
Conversely, assume that Dfu+Dfv is an arithmetic progression. Then, by previous step, we have Dfu+Dfv=DA, where A=fu+fv. Then, we have DA is an arithmetic progression. Since the difference set DA is an arithmetic progression, then by the above remark, we have A=fu+fv=f+uv is also an arithmetic progression. Hence, the edge e=uv has an arithmetic progression as its set-label. □
In view of the notions mentioned above, we note that there are some graphs, all whose elements have arithmetic progressions as their set-label and there are some graphs, the set-label of whose edges are not arithmetic progressions. Keeping this in mind, we define the following notion.
Definition 3.2. An arithmetic sumset labelling of a graph G is a modular sumset labelling f of G, with respect to which the set-label of all vertices and edges of G are arithmetic progressions. A graph that admits an arithmetic modular sumset labelling is called an arithmetic modular sumset graph.
Analogous to the condition for an arithmetic sumset graphs (see [44]), a necessary and sufficient condition for a graph to admit an arithmetic modular sumset labelling is discussed in the following theorem.
Theorem 3.2.A graph G admits an arithmetic modular sumset labelling f if and only if for any two adjacent vertices in G, the deterministic ratio of every edge of G is a positive integer, which is less than or equal to the set-labelling number of its end vertex having smaller common difference.
Proof. Here, we need to consider the following two cases:
Case 1: First note that if the set-label of two adjacent vertices are arithmetic progressions with the same common difference, say d, then the set-label of the corresponding edge is also an arithmetic progression with the same common difference d. Then, it is clear that a vertex arithmetic modular sumset graph is an arithmetic modular sumset graph if the common differences between any two adjacent vertices of G are the same.
Case 2: Assume that u,v be any two adjacent vertices in G with common differences du and dv respectively such that du≤dv. Also, assume that fu=ar=a+rdu:0≤r<m and fv=bs=b+sdv:0≤s<n. Then, ∣fu∣=m and ∣fv∣=n. Now, arrange the terms of fu+fv=f+uv in rows and columns as follows. For any bs∈fv,0≤s<n, arrange the terms of A+bs in s+1th row in such a way that equal terms of different rows come in the same column of this arrangement. Without loss of generality, assume that dv=kdu and k≤m. If k<m, then for any a∈fu and b∈fv we have a+b+dv=a+b+kdu<a+b+mdi. That is, a few final elements of each row of the above arrangement occur as the initial elements of the succeeding row (or rows) and the difference between two successive elements in each row is du itself. If k=m, then the difference between the final element of each row and the first element of the next row is du and the difference between two consecutive elements in each row is du. Hence, if k≤m, then f+uv is an arithmetic progression with common difference du.
In both cases, note that if the given conditions are true, then f is an arithmetic modular sumset labelling of G.
We prove the converse part by contradiction method. For this, assume that f is an arithmetic modular sumset labelling of G. Let us proceed by considering the following two cases.
Case-1: Assume that dj is not a multiple of di (or di is not a multiple of dj). Without loss generality, let di<dj. Then, by division algorithm, dj=pdi+q,0<q<di. Then, the differences between any two consecutive terms in f+vivj are not equal. Hence, in this case also, f is not an arithmetic modular sumset labelling, contradiction to the hypothesis. Therefore, di∣dj.
Case 2: Let dj=kdi where k>m. Then, the difference between two successive elements in each row is di, but the difference between the final element of each row and the first element of the next row is tdi, where t=k−m+1≠1. Hence, f is not an arithmetic modular sumset labelling, a contradiction to the hypothesis. Hence, we have dj=kdi;k≤m.
Therefore, from the above cases it can be noted that if a vertex arithmetic modular sumset labelling of G is an arithmetic modular sumset labelling of G, then the deterministic ratio of every edge of G is a positive integer, which is greater than or equal to the set-labelling number of its end vertex having smaller common difference. This completes the proof. □
In the following theorem, we establish a relation between the common differences of the elements of an arithmetic modular sumset graph G.
Theorem 3.3.If G is an arithmetic modular sumset graph, the greatest common divisor of the common differences of vertices of G and the greatest common divisor of the common differences of the edges of G are equal to the smallest among the common differences of the vertices of G.
Proof. Let f be an arithmetic modular sumset labelling of G. Then, by Theorem 3.2, for any two adjacent vertices vi and vj of G with common differences di and dj respectively, either di=dj, or if dj>di,dj=kdj, where k is a positive integer such that 1<k≤∣fvi∣.
If the common differences of the elements of G are the same, the result is obvious. Hence, assume that for any two adjacent vertices vi and vj of G, dj=kdj,k≤∣fvi∣, where di is the smallest among the common differences of the vertices of G. If vr is another vertex that is adjacent to vj, then it has the common difference dr which is equal to either di or dj or ldj. In all the three cases, dr is a multiple of di. Hence, the greatest common divisor of di,dj,dr is di. Proceeding like this, we have the greatest common divisor of the common differences of the vertices of G is di.
Also, by Theorem 3.2, the edge uivj has the common difference di. The edge vjvk has the common difference di, if dk=di, or dj in the other two cases. Proceeding like this, we observe that the greatest common divisor of the common differences of the edges of G is also di. This completes the proof.□
The study on the set-labelling number of edges of an arithmetic modular sumset graphs arouses much interest. Analogous to the result on set-labelling number of the edges of an arithmetic sumset graph (see [43]), The set-labelling number of an edge of an arithmetic modular sumset graph G, in terms of the set-labelling numbers of its end vertices, is determined in the following theorem.
Theorem 3.4.Let G be a graph which admits an arithmetic modular sumset labelling, say f and letdianddjbe the common differences of two adjacent verticesviandvjin G. If∣fvi∣≥∣fvj∣, then for some positive integer1≤k≤∣fvi∣, the edgevivjhas the set-labelling number∣fvi∣+kfvj−1.
Proof. Let f be an arithmetic modular sumset labelling defined on G. For any two vertices vi and vj of G, let fvi=aiai+diai+2diai+3di…ai+m−1di and let fvj=ajaj+djaj+2djaj+3dj…aj+n−1dj. Here ∣fvi∣=m and ∣fvj∣=n.
Let di and dj be the common differences of the vertices vi and vj respectively, such that di<dj. Since f is an arithmetic modular sumset labelling on G, by Theorem 3.2, there exists a positive integer k such that dj=k.di, where 1≤k≤∣fvi∣. Then, fvj=ajaj+kdiaj+2kdiaj+3kdi…aj+n−1kdi. Therefore, f+vivj=ai+ajai+aj+diai+aj+2di…ai+aj+m−1+kn−1di. That is, the set-labelling number of the edge vivj is m+kn−1. □
4. Strongly modular sumset graphs
The next type of a modular sumset labelling we are going to discuss is the one with the upper bound in Inequality (1) is sharp (that is, ∣A+B∣=∣A‖B∣). Thus, we have the following definition.
Definition 4.1. [47] A modular sumset labelling f:VG→P0Zn defined on a given graph G is said to be a strongly modular sumset labelling if for the associated function f+:EG→P0Zn, ∣f+uv∣=∣fu∣∣fv∣∀uv∈EG. A graph which admits a strongly modular sumset labelling is called a strongly modular sumset graph.
Invoking the notion difference set of a set, a necessary and sufficient condition of a modular sumset labelling of a graph G to be a strongly modular sumset labelling is given below:
Theorem 4.1.A modular sumset labellingf:VG→P0Znof a given graph G is a strongly modular sumset labelling of G if and only ifDfu∩Dfv=Ø,∀uv∈EG, where∣fu∣∣fv∣≤n.
Proof. Let f:VG→P0Zn be a modular sumset labelling on a given graph G. For any vertex u∈VG, define Dfu=ai−aj:aiaj∈fu.
Let uv be an arbitrary edge in EG. Assume that f is a strong modular sumset labelling of G. Then, by definition ∣f+uv∣=∣fu∣∣fv∣. Therefore, for any elements ai,aj∈fu and br,bs∈fv, we have ai+br≠aj+bs in f+uv∀uv∈EG. That is, ∣ai−aj∣≠∣bs−br∣ for any ai,aj∈fu and br,bs∈fv. That is, Dfu∩Dfv=Ø. Therefore, the difference sets of the set-label of any two adjacent vertices are disjoint.
Conversely, assume that the difference Dfu∩Dfv=Ø for any edge uv in G. That is, ∣ai−aj∣≠∣bs−br∣ for any ai,aj∈fu and br,bs∈fv. Then, ai−aj≠bs−br. That is, ai+br≠aj+bs. Therefore, all elements in fu+fv are distinct. That is, ∣f+uv∣=∣fu∣∣fv∣ for any edge uv∈EG. Hence, f is a strongly modular sumset labelling of G.
Also, note that the maximum possible cardinality in the set-label of any element of G is n, the product ∣fu∣∣fv∣ cannot exceed the number n. This completes the proof. □
A necessary and sufficient condition for a modular sumset labelling of a graph G to be a strongly k-uniform modular sumset labelling is given below:
Theorem 4.2. [47] For a positive integerk≤n, a modular sumset labellingf:VG→P0Znof a given connected graph G is a stronglyk-uniform modular sumset labelling of G if and only if eitherkis a perfect square or G is bipartite.
Proof. If k is a perfect square, say k=l2, then we can label all the vertices of a graph by distinct l-element sets in such a way that the difference sets of the set-label of every pair of adjacent vertices are disjoint. Hence, assume that k is not a perfect square.
Let G be a bipartite graph with bipartition XY. Let r,s be two divisors of k. Label all vertices of X by distinct r-element sets all of whose difference sets are the same, say DX. Similarly, label all vertices of Y by distinct s-element sets all of whose difference sets the same, say DY, such that DX∩DY=Ø. Then, all the edges of G have the set-labelling number k=rs. Therefore, G is a strongly k-uniform modular sumset graph.
Conversely, assume that G admits a strongly k-uniform modular sumset labelling, say f. Then, f+uv=k∀uv∈EG. Since, f is a strong modular sumset labelling, the set-labelling number of every vertex of G is a divisor of the set-labelling numbers of the edges incident on that vertex. Let v be a vertex of G with the set-labelling number r, where r is a divisor of k, but r2≠k. Since f is k-uniform, all the vertices in Nv, must have the set-labelling number s, where rs=k. Again, all vertices, which are adjacent to the vertices of Nv, must have the set-labelling number r. Since G is a connected graph, all vertices of G have the set-labelling number r or s. Let X be the set of all vertices of G having the set-labelling number r and Y be the set of all vertices of G having the set-labelling number s. Since r2≠k, no two elements in X (and no elements in Y also) can be adjacent to each other. Therefore, G is bipartite.□
The following result is an immediate consequence of the above theorem.
Theorem 4.3. [47] For a positive non-square integerk≤n, a modular sumset labellingf:VG→P0Znof an arbitrary graph G is a stronglyk-uniform modular sumset labelling of G if and only if either G is bipartite or a disjoint union of bipartite components.
For a positive integer k≤n, the maximum number of components in a strongly k-uniform modular sumset graph is as follows.
Proposition 4.4. [47] Let f be a stronglyk-uniform modular sumset labelling of a graph G with respect to the ground setZn. Then, the maximum number of components in G is the number of distinct pairs of divisorsrandsofksuch thatrs=k.
The following theorem discusses the condition for an arithmetic modular sumset labelling of a graph G to be a strongly modular sumset labelling of the graph.
Theorem 4.5.LetGbe a graph which admits an arithmetic modular sumset labelling, sayf. Then,fis a strongly modular sumset labelling ofGif and only if the deterministic ratio of every edge ofGis equal to the set-labelling number of its end vertex having smaller common difference.
Proof. Let f be an arithmetic modular sumset labelling of G. Let vi and vj are two adjacent vertices in G and di and dj be their common differences under f. Without loss of generality, let di<dj. Then, by Theorem 3.4, the set-labelling number of the edge vivj is ∣fvi∣+kfvj−1.
Assume that f is a strongly modular sumset labelling. Therefore, f+vivj=mn. Then,
Conversely, assume that the common differences di and dj of two adjacent vertices vi and vj respectively in G, where di<dj such that dj=∣fvi∣.di. Assume that fvi=ar=a+rdi:0≤r<fvi and fvj=bs=b+skdi:0≤s<fvj, where k≤∣fvi∣. Now, arrange the terms of f+vivj=fvi+fvj in rows and columns as follows. For bs∈fvj,0≤s<∣fvj∣, arrange the terms of fvi+bs in s+1-th row in such a way that equal terms of different rows come in the same column of this arrangement. Then, the common difference between consecutive elements in each row is di. Since k=∣fvi∣, the difference between the final element of any row (other than the last row) and first element of its succeeding row is also di. That is, no column in this arrangement contains more than one element. Hence, all elements in this arrangement are distinct. Therefore, total number of elements in fvi+fvj is ∣fvi∣∣fvj∣. Hence, f is a strongly modular sumset labelling.□
5. Supreme modular sumset labelling of G
In both types of modular sumset labelling discussed above, it is observed that the cardinality of the edge set-label cannot exceed the value n. This fact creates much interest in investigating the case where all the edge set-label have the cardinality n.
Definition 5.1. [47] A modular sumset labelling f:VG→P0Zn of a given graph G is said to be a supreme modular sumset labelling or maximal modular sumset labelling of G if and only if f+EG=Zn.
Put in a different way, a modular sumset labelling f:VG→PZn of a given graph G is a supreme modular sumset labelling of G if the set-label of every edge of G is the ground set Zn itself.
A necessary and sufficient condition for a modular sumset labelling of a graph G to be its supreme modular sumset labelling is discussed in the theorem given below:
Theorem 5.1. [47] The modular sumset labellingf:VG→PZnof a given graphGis a supreme modular sumset labelling ofGif and only if for every pair of adjacent verticesuandvofGsome or all of the following conditions hold.
∣fu∣+∣fv∣≥n if Dfu∩Dfv≠Ø. The strict inequality hold when Dfu and Dfv are arithmetic progressions containing the same elements.
∣fu∣∣fv∣≥n if Dfu∩Dfv=Ø.
Proof. For two adjacent vertices u and v in G, let Dfu=Dfv=d are arithmetic progressions containing the same elements. Then, the elements in fu and fv are also in arithmetic progression, with the same common difference d. Then, by Theorem 3.4, n=∣fu+fv∣=∣fu∣+∣fv∣−1. Therefore, the set-labelling number of the edge uv is n if and only if ∣fu∣+∣fv∣>n.
Now, let Dfu∩Dfv≠Ø such that Dfu≠Dfv. Then, clearly ∣fu+fv∣≥∣fu∣+∣fv∣. Therefore, we have ∣f+uv∣=n if and only if ∣fu∣+∣fv∣≥n.□
Next assume that Dfu∩Dfv=Ø. Then, ∣fu+fv∣=∣fu∣∣fv∣. Therefore, we have ∣f+uv∣=n if and only if ∣fu∣∣fv∣≥n.
A necessary and sufficient condition for a strong modular sumset labelling of a graph G to be a maximal modular sumset labelling of G.
Theorem 5.2. [47] Letfbe a strong sumset-labelling of a given graphG. Then,fis a maximal sumset-labelling ofGif and only ifnis a perfect square orGis bipartite or a disjoint union of bipartite components.
Proof. The proof is an immediate consequence of Theorem 4.2, when k=n.□
6. Weakly modular sumset graphs
Another interesting question we address in the beginning of this section is whether the lower bound and the upper bound of the sumset can be equal. Suppose that A and B be two non-empty subsets of Zn such that the bounds of their sumset are equal. Then, we have
which is possible only when ∣A∣=1 or ∣B∣=1 (or both). Also, note that in this case the cardinality of the sumset is equal to equal to that of one of the summands. This interesting phenomenon leads us to a new type of a modular sumset labelling called weakly modular sumset labelling. This type of labelling is investigated in the following section.
6.1 Weakly modular sumset labelling of graphs
Definition 6.1. A modular sumset labelling f of a graph G is said to be a weakly modular sumset labelling of G if the cardinality of the set-label of every edge of G is equal to the cardinality of the set-label of at least one of its end vertices. A graph which admits a weakly modular sumset labelling is called a weakly modular sumset graph.
From the above definition, it can be observed that for any edge uv in weakly modular sumset graph G, ∣f+uv∣=∣fu∣ or ∣f+uv∣=∣fv∣. Putting it in a different way, the set-labelling number of at least one end vertex of every edge of a weakly modular sumset graph is a singleton. An element (a vertex or an edge) of modular sumset graph G with set-labelling number 1 is called a sparing element or a monocardinal elements of G. Hence, analogous to the condition for a sumset graph to be a weak sumset graph (see [39]), we have.
Theorem 6.1.A graphGadmits a weak modular sumset labelling if and only ifGis bipartite or contains sparing edges.
Proof. Note the fact that at least one end vertex of every edge of G is a sparing vertex. Also, we note that no two vertices with non-singleton set-label in weakly modular sumset graph can be adjacent to each other. Thus, if every of edge of G has exactly one end vertex with singleton set-label, then we can partition the vertex set of G into two subsets X with all sparing vertices and Y with all non-sparing vertices. Here, no two vertices in the same partition are adjacent and hence G is a bipartite graph. If G is not a bipartite graph, then obviously G should have at least one sparing edge, completing the proof.
Invoking Theorem 6.1, the following two results are immediate.
Corollary 6.2.Every graphGadmits a weakly modular sumset labelling.
Corollary 6.3.A graphGadmits a weakly uniform modular sumset labelling if and only ifGis bipartite.
The above results are similar to the corresponding result of integer additive set-labelled graphs (see [39]) and hence the notion of sparing number of graphs defined and studied in [48, 49, 50, 51, 52, 53, 54, 55, 56, 57] can be extended to our current discussion also. The notion of the sparing number of graphs is defined as follows:
Definition 6.2. Let G be a weakly modular sumset labelled graph. Then, the sparing number of G is the number of sparing edges in G.
A set of vertices X of a graph G is said to have maximal incidence if the maximum number of edges of incidence at the elements of X. Then, analogous to the corresponding result of integer additive set-valued graphs (see [40]), we have.
Theorem 6.4.LetGbe a weakly modular sumset labelled graph andIbeIbe the largest independent set ofGwith maximum incidence. Then, the sparing number ofGis∣EG∣−∑vi∈Idvi.
Proof. Recall that the degree of a vertex v, denoted by dv, is equal to the number of edges incident on a vertex. Note that any vertex vi∈I can have a non-singleton set-label which gives non-singleton set labels to dvi edges incident on it. Since I is an independent set, the edges incident at the vertices in I assumes non-singleton set-label. Therefore, the number of edges having non-singleton set-label incident at the vertices in I is ∑vi∈Idvi. Since I is a maximal independent set of that kind, the above expression counts the maximal non-sparing edges in G. Hence, the number of sparing edges in G is ∣EG∣−∑vi∈Idvi. □
6.2 Weakly modular sumset number of graphs
As a special case of the modular sumset number, the notion of weakly modular sumset number is introduced in [47] as follows:
Definition 6.3. The weakly modular sumset number of a graph G, denoted by σw is defined to be the minimum value of n such that a modular sumset labelling f:VG→P0Zn is a weakly modular sumset labelling of G.
The following theorem discussed the weak sumset number of an arbitrary graph G in terms of its covering and independence numbers.
Theorem 6.5. [47] LetGbe a modular sumset graph andαandβbe the covering number and independence number ofGrespectively. Then, the weak modular sumset number ofGismaxαr, whereris the smallest positive integer such that2r−r−1≥β.
Proof. Recall that αG+βG=∣VG∣ (see [4]). Since G is a modular sumset graph, no two adjacent vertices can have non-singleton set-label simultaneously. Therefore, the maximum number of vertices that have non-singleton set-label is β. Let V′ be the set of these independent vertices in G. Therefore, the minimum number of sparing vertices is ∣VG∣−β=α. Since all these vertices in V−V′ must have distinct singleton set-label, the ground set must have at least α elements.
Also, the number non-empty, non-singleton subsets of the ground set must be greater than or equal to α. Otherwise, all the vertices in V′ cannot be labelled by non-singleton subsets of this ground set. We know that the number of non-empty, non-singleton subsets of a set A is 2∣A∣−∣A∣−1, where A⊆Zn is the ground set used for labelling.
Therefore, the weak modular sumset number G is α if 2α−α−1≥β. Otherwise, the ground set must have at least r elements such that 2r−r−1≥β. Therefore, in this case, the weak modular sumset number of G is r, where r is the smallest positive integer such that 2r−r−1≥β. Hence, σ∗G=maxαr. This completes the proof.□
The weakly modular sumset number some fundamental graph classes are given in Table 1.
Graph class
σ*(G)
Path, Pp
2 if p≤2;⌊p2⌋ if p>2
Cycle, Cp
p−1 if p=3,4; ⌊p2⌋ if p>4
Wheel graph, W1,p
1 + ⌊p2⌋
Helm graph, Hp
p
Ladder graph, Lp
p
Complete graph, Kp
p−1
Table 1.
Weakly modular sumset number of some graph classes.
The following theorem discusses the minimum cardinality of the ground set when the given graph G admits a weakly uniform modular sumset labelling.
Theorem 6.6. [47] LetGbe a weaklyk-uniform modular sumset graph with covering numberαand independence numberβ, wherek<α. Then, the minimum cardinality of the ground setZnismaxαr, where r is the smallest positive integer such thatrk≥β.
Proof. Let a weakly k-uniform modular sumset labelling be defined on a graph G over the ground set A⊂Zn. Then, by Corollary 6.3, G is bipartite. Let X,Y be the bipartition of the vertex set VG. Without loss of generality, let ∣X∣≤∣Y∣. Then, α=∣X∣ and β=∣Y∣. Then, distinct elements of X must have distinct singleton set-label. Therefore, n≥α.
On the other hand, since f is k-uniform, all the elements in Y must have distinct k-element set-label. The number of k-element subsets of a set A (obviously, with more than k elements) is ∣A∣k. The ground set A has α elements only if αk≥β. Otherwise, the ground set A must contain at least r elements, where r>α is the smallest positive integer such that rk≥β. Therefore, n=maxαr.□
In view of the above theorem, the following result is immediate.
Corollary 6.7.LetGbe a weaklyk-uniform modular sumset graph, wherek≥α, whereαis the covering number ofG. Then, the minimum cardinality of the ground set \tZnis the smallest positive integernsuch thatnk≥β, whereβis the independence number ofG.
The following result explains a necessary and sufficient condition for a weak modular sumset labelling of a given graph G to be a maximal modular sumset labelling of G.
Proposition 6.8. [47] A weakly modular sumset labelling of a graphGis a supreme modular sumset labelling ofGif and only ifGis a star graph.
Proof. Let f be a weak modular sumset labelling of given graph G. First, assume that f is a maximal modular sumset labelling of G. Then, the set-labelling number of one end vertex of every edge of G is 1 and the set-labelling number of the other end vertex is n. Therefore, Zn be the set-label of one end vertex of every edge of G, which is possible only if G is a star graph with the central vertex has the set-label Zn and the pendant vertices of G have distinct singleton set-label.
Conversely, assume that G is a star graph. Label the central vertex of G by the ground set Zn and label other vertices of G by distinct singleton subsets of Zn. Then, all the edges of G has the set-labelling number n. That is, this labelling is a supreme modular sumset labelling of G.□
7. Conclusion
In this chapter, we have discussed certain types of modular sumset graphs and their structural properties and characterisations. These studies are based on the cardinality of the set-label of the elements of the graphs concerned and the patterns of the elements in these set-label. It is to be noted that several other possibilities can be investigated in this regard. For example, analogous to the topological set-valuations of graphs, the case when the collection of set-label of vertices and/or edges of a graph G forms a topology of the ground set Zn can be studied in detail. Another possibility for future investigation is to extend the graceful and sequential concepts of set-labelling of graphs to modular sumset labelling also. All these points highlight the wide scope for further studies in this area.
Acknowledgments
The author would like to dedicate the chapter to his doctoral thesis supervisor Prof. Dr. Germina K Augusthy for her constant support and inspiration for working in the area of sumset-labelling. The author would also like to acknowledge the critical and creative suggestions of the editor and referees which improved the content and the presentation style of the chapter.
Additional information
Mathematics Subject Classification 2010: 05C75
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Introduction",level:"1"},{id:"sec_1_2",title:"1.1 Basics of graph labelling",level:"2"},{id:"sec_2_2",title:"1.2 Sumsets and integer additive set-labelled graphs",level:"2"},{id:"sec_4",title:"2. Modular sumset labelling of graphs",level:"1"},{id:"sec_4_2",title:"2.1 Basics of modular sumsets",level:"2"},{id:"sec_5_2",title:"2.2 Modular sumset graphs",level:"2"},{id:"sec_7",title:"3. Arithmetic modular sumset graphs",level:"1"},{id:"sec_8",title:"4. Strongly modular sumset graphs",level:"1"},{id:"sec_9",title:"5. Supreme modular sumset labelling of G",level:"1"},{id:"sec_10",title:"6. Weakly modular sumset graphs",level:"1"},{id:"sec_10_2",title:"6.1 Weakly modular sumset labelling of graphs",level:"2"},{id:"sec_11_2",title:"6.2 Weakly modular sumset number of graphs",level:"2"},{id:"sec_13",title:"7. Conclusion",level:"1"},{id:"sec_14",title:"Acknowledgments",level:"1"},{id:"sec_14",title:"Additional information",level:"1"}],chapterReferences:[{id:"B1",body:'Bondy JA, Murty USR. 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Journal of Discrete Mathematical Sciences & Cryptography. 2015;18(1–2):117-128'},{id:"B42",body:'Sudev NK, Germina KA. Some new results on strong integer additive set-indexers of graphs. Discrete Mathematics, Algorithms and Applications. 2015;7(01):1450065'},{id:"B43",body:'Sudev NK, Germina KA. On certain arithmetic integer additive set-indexers of graphs. Discrete Mathematics, Algorithms and Applications. 2015;7(03):1550025'},{id:"B44",body:'Sudev NK, Germina KA. A study on arithmetic integer additive set-indexers of graphs. Journal of Informatics and Mathematical Sciences. 2018;10(1–2):321-332'},{id:"B45",body:'Naduvath S, Augusthy GK, Kok J. Sumset valuations of graphs and their applications. In: Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey, Pennsylvania: IGI Global; 2020. pp. 208-250'},{id:"B46",body:'Naduvath S, Germina KA. An Introduction to Sumset Valued Graphs. Mauritius: Lambert Academic Publ; 2018'},{id:"B47",body:'Naduvath S. A study on the modular sumset labeling of graphs. Discrete Mathematics, Algorithms and Applications. 2017;9(03):1750039'},{id:"B48",body:'Chithra KP, Sudev NK, Germina KA. Sparing number of Cartesian products of certain graphs. Communications in Mathematics and Applications. 2014;5(1):23-30'},{id:"B49",body:'Chithra KP, Sudev NK, Germina KA. A study on the sparing number of corona of certain graphs. Research & Reviews: Discrete Mathematical Structures. 2014;1(2):5-15'},{id:"B50",body:'Naduvath S, Kaithavalappil C, Augustine G. A note on the sparing number of generalised petersen graphs. Journal of Combinatorics, Information & System Sciences. 2017;42(1–2):23-31'},{id:"B51",body:'Sudev NK, Germina KA. A note on the sparing number of graphs. Advances and Applications in Discrete Mathematics. 2014;14(1):51-65'},{id:"B52",body:'Sudev NK, Germina KA. On the sparing number of certain graph structures. 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