Research article Special Issues

Skewness and the crossing numbers of graphs

  • Received: 28 March 2023 Revised: 12 July 2023 Accepted: 21 July 2023 Published: 07 August 2023
  • MSC : 05C10, 05C62

  • The skewness of a graph G, sk(G), is the smallest number of edges that need to be removed from G to make it planar. The crossing number of a graph G, cr(G), is the minimum number of crossings over all possible drawings of G. There is minimal work concerning the relationship between skewness and crossing numbers. In this work, we first introduce an inequality relation for these two parameters, and then we construct infinitely many near-planar graphs such that the inequality is equal. In addition, we give a necessary and sufficient condition for a graph to have its skewness equal to the crossing number and characterize some special graphs with sk(G)=cr(G).

    Citation: Zongpeng Ding. Skewness and the crossing numbers of graphs[J]. AIMS Mathematics, 2023, 8(10): 23989-23996. doi: 10.3934/math.20231223

    Related Papers:

    [1] Lamya Almaghamsi, Aeshah Alghamdi, Abdeljabbar Ghanmi . Existence of solution for a Langevin equation involving the ψ-Hilfer fractional derivative: A variational approach. AIMS Mathematics, 2025, 10(1): 534-550. doi: 10.3934/math.2025024
    [2] Arjumand Seemab, Mujeeb ur Rehman, Jehad Alzabut, Yassine Adjabi, Mohammed S. Abdo . Langevin equation with nonlocal boundary conditions involving a ψ-Caputo fractional operators of different orders. AIMS Mathematics, 2021, 6(7): 6749-6780. doi: 10.3934/math.2021397
    [3] Sotiris K. Ntouyas, Bashir Ahmad, Jessada Tariboon . Nonlocal integro-multistrip-multipoint boundary value problems for ¯ψ-Hilfer proportional fractional differential equations and inclusions. AIMS Mathematics, 2023, 8(6): 14086-14110. doi: 10.3934/math.2023720
    [4] Karim Guida, Lahcen Ibnelazyz, Khalid Hilal, Said Melliani . Existence and uniqueness results for sequential ψ-Hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(8): 8239-8255. doi: 10.3934/math.2021477
    [5] Sunisa Theswan, Sotiris K. Ntouyas, Jessada Tariboon . Coupled systems of ψ-Hilfer generalized proportional fractional nonlocal mixed boundary value problems. AIMS Mathematics, 2023, 8(9): 22009-22036. doi: 10.3934/math.20231122
    [6] Thabet Abdeljawad, Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Eman Al-Sarairah, Artion Kashuri, Kamsing Nonlaopon . Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application. AIMS Mathematics, 2023, 8(2): 3469-3483. doi: 10.3934/math.2023177
    [7] Zohreh Heydarpour, Maryam Naderi Parizi, Rahimeh Ghorbnian, Mehran Ghaderi, Shahram Rezapour, Amir Mosavi . A study on a special case of the Sturm-Liouville equation using the Mittag-Leffler function and a new type of contraction. AIMS Mathematics, 2022, 7(10): 18253-18279. doi: 10.3934/math.20221004
    [8] Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for ψ-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244
    [9] Zheng Kou, Saeed Kosari . On a generalization of fractional Langevin equation with boundary conditions. AIMS Mathematics, 2022, 7(1): 1333-1345. doi: 10.3934/math.2022079
    [10] Muath Awadalla, Manigandan Murugesan, Subramanian Muthaiah, Bundit Unyong, Ria H Egami . Existence results for a system of sequential differential equations with varying fractional orders via Hilfer-Hadamard sense. AIMS Mathematics, 2024, 9(4): 9926-9950. doi: 10.3934/math.2024486
  • The skewness of a graph G, sk(G), is the smallest number of edges that need to be removed from G to make it planar. The crossing number of a graph G, cr(G), is the minimum number of crossings over all possible drawings of G. There is minimal work concerning the relationship between skewness and crossing numbers. In this work, we first introduce an inequality relation for these two parameters, and then we construct infinitely many near-planar graphs such that the inequality is equal. In addition, we give a necessary and sufficient condition for a graph to have its skewness equal to the crossing number and characterize some special graphs with sk(G)=cr(G).



    There are many life-threatening diseases, out of which Malaria is one of them that is produced through the parasite known as Plasmodium that transmits to humans with the infected anopheles mosquito bite [1,2,3]. People with malaria are infected by the Protozoan parasite, which grows within the cells of red blood and spreads the illness. There are four Plasmodia species that can become the reason of infection, known as oval, falciparum, malaria, and vivax, while the two other most widespread Plasmodium species are falciparum and vivax [4]. Malaria affects lots of people each year, mainly in South Asia, Africa, and South America. It becomes the major reason of death and illness in various developing regions, specifically young children, and pregnant women. The main symptoms of malaria are fever, muscle pain, and headache. Malaria can progress promptly, possibly severe anemia, lung problems, and even cause death. There are various effective tactics that proposed the World Health Organization (WHO) to treat and prevent this dangerous disease. Some of them are indoor spraying, pesticide-treated bed nets, and some antimalarial medications. The vector control systems can be applied to lessen the poisoned mosquito population. In general, malaria is a substantial global health task, which needs a comprehensive approach for its cure by using the strategies of vector control and early identification for its treatment.

    Malaria is a dynamic and complicated disease, which is considered difficult to present its model mathematically. The malaria parasite's life cycle is complex due to numerous phases of development both in mosquito vector and human host. These points are prompted by numerous features, e.g., mosquito actions, host immunity, and environmental environments that is used to make difficult the disease dynamics to predict and understand. The intricacy of the virus's spread dynamics is one of the main problems with malaria modelling. Malaria spread is influenced by numerous interrelated influences, e.g., the mosquito vector density, the host immunity degree, and the accessibility of efficient mediations. The relations between these aspects indicate the nonlinearity and hard to capture using the mathematical systems [5]. However, the mathematical malaria disease model based on the pesticides and medication (MDMPM) is presented, which is categorized in four human (susceptible, exposed, infected, recovered) and two vector (susceptible, infected) classes. The pathogen infects the population of host then the individuals are categorized into groups based on the parasites number and virus types. Kermack-McKendrick [6] proposed the mathematical SEIRSI model based on the population of human/vector groups. The nonlinear dynamics of the SEIRSI model is presented as [7]:

    Sh(x)=αh+β1Rh(x)γ1ψIh(x)Sh(x)γ2ξIm(x)Sh(x)(δ1+ηh)Sh(x),Eh(x)=γ2ξIm(x)Sh(x)+γ1ψIh(x)Sh(x)(β2+δ2+ηh)Eh(x),Ih(x)=β2Eh(x)(ϵ+pυ+ηh)Ih(x),Rh(x)=δ1Sh(x)+υpIh(x)+δ2Eh(x)(β1+ηh)Rh(x),Sm(x)=αm(σ+ηm)Sm(x)γ3eSm(x)Ih(x),Im(x)=γ3eIh(x)Sm(x)(σ+ηm)Im(x). (1)

    The compartmental illustration of the SEIRSI model based on the MDMPM is shown in Figure 1, while the description of each parameter is given in Table 1.

    Figure 1.  Graphical representations of the MDMPM.
    Table 1.  Parameter details of the MDMPM.
    Parameters Details
    β1 Susceptible form of recovered individuals
    αm Newborn mosquitoes transfer rate into Sm
    αh Newborns ratio to enter in Sh
    γ1 Transfer ratio of illness from one affected to susceptible people
    ψ Transfusions rate of average blood with stable time in Ih and Sh
    β2 Exposed to infected infection rate in people
    ξ Infected mosquito susceptible bites using fixed time
    γ3 Rate of Mosquitos infection
    γ2 Disease transfer through infected (mosquito) to susceptible (individual)
    ηh People natural rate of death
    δ2 Exposed recovered population
    δ1 Susceptible recovered population
    p Rate of humans heal
    ε Infected human death due to disease
    ηm Mosquitos' natural death
    υ Anti-malarial prescriptions
    e Bites Sm using the infected humans
    σ Expires sections using spraying for Sm and Im
    x Time

     | Show Table
    DownLoad: CSV

    There are various deterministic and conventional numerical schemes that have been presented to solve the biological models, including the wavelet Bernstein scheme that has been presented to solve the epidemic SIR fractional kind of infectious disease model [8], Laplace method is applied for the fractional epidemic system of born vector disease [9], q-homotopy transform technique is proposed to solve the fractional Schistosomiasis disease model [10,11], a theoretic investigation is presented based on the impacts of vector-virus transmission mechanism [12], a collocation technique with shifted Chebyshev polynomials is applied to solve the fractional order diseased model of Rubella ailment [13], an analytical computational approach is provided for the inflammatory atherosclerosis disease system [14], the differential stomach and coronavirus models have been discussed numerically by applying the Adam method [15,16] and the design of a numerical method is presented for the maturing communication in population and epidemic models [17]. Some more deterministic procedures that have been applied to solve different dynamical models are presented in [18,19,20,21,22].

    All mentioned schemes have their individual strength, and demerits, however the stochastic solvers have never been tested before for the nonlinear dynamics of the MDMPM. The current study is related to the numerical solutions of the MDMPM through the stochastic procedures using the scale conjugate gradient neural networks (SCJGNNs). In recent years, the stochastic computing paradigms have been proposed in various applications, e.g., eye surgery differential systems [23,24], food chain systems [25,26], smoking dynamical systems [27], models arising in biological applications, [28,29], thermal explosion singular systems [30,31], and many more [32,33,34,35,36,37,38,39,40,41,42,43]. Based on these noteworthy applications of neuro-computing paradigm, the authors are motivated to present the numerical solutions of the malaria disease model of pesticides and medication (MDMPM) with stochastic framework of scale conjugate gradient neural networks (SCJGNNs). The purpose of this work is to provide a stochastic framework based on the SCJGNNs for solving the mathematical model representing MDMPM dynamics.

    The contributions, insights and innovative aspects of the current scheme is portrayed in terms of salient features as follows:

    • A novel design of the stochastic framework via SCJGNNs is introduced/presented to get the numerical results/outcomes of the nonlinear dynamics of governing mathematical model of MDMPM.

    • The capability and knacks of Runge-Kutta solver is exploited/developed to formulate the target datasets for different scenario of MDMPM system that are used for the testing, and training procedure of the networks via SCJGNNs.

    • The exactness of the stochastic SCJGNNs is attained/verified/validated through the overlapping of the results with standard counterparts having very small magnitude of absolute error (AE).

    • The reliable matching of the numerical results, a strong arrangement using the benchmark outputs along with consistent statistical performance indices further prove the importance/significance/role of the SCJGNNs to solve the variant of MDMPM models.

    The remaining paper is presented as: Proposed SCJGNNs design is shown in Section 2, Sect 3 presents the numerical results, and the conclusions are discussed in the final section.

    The current section is used to design the computing SCJGNNs structure to solve the MDMPM in two steps, including the detail of designed SCJGNNs performances along with its execution. The designed SCJGNNs is presented in Figure 2. In the 1st phase of Figure 2, the nonlinear mathematical system is illustrated, the stochastic methodology using the SCJGNNs is provided in the 2nd phase of Figure 2, while some results are presented in the last phase of Figure 2. The performances using the multi-layers network based on the structure of single neuron is illustrated in Figure 3.

    Figure 2.  Illustration of proposed SCJGNNs performances to solve the MDMPM.
    Figure 3.  Single layer neural network.

    The computing SCJGNNs approach for solving the MDMPM is executed using the command 'nftool' in software 'MATLAB', n-folded cross authentication, layers (output/hidden/input) structure, epochs are taken maximum 1000, fifteen hidden neurons, transfer function log-sigmoid, stopping standards, SCJG algorithm, tolerances, and step size. The label statics and input/target training are achieved using the basic outputs by taking the statics of validation (12%), testing (10%), and training (78%) for the MDMPM.

    The designed computing SCJGNNs for the solutions of the MDMPM is performed by using 15 neurons, which have been validated through the best values of underfitting and overfitting based on the authorization and training at epochs 100, 86 and 69. For small neurons, the premature convergence (under fitting) problem is faced, while similar precision by using higher intricacy (overfitting issue) is noticed by taking higher neurons. The trials have been taken for the validation (12%), testing (10%), and training (78%) to solve the MDMPM. By taking a smaller data training, like 40% to 60%, the MSE values for testing 30% to 40% increase and generally poor presentations have been observed. On the other hand, if the training statics is almost designated > 90% and 5% for both testing and certification, the procedure's correctness improves marginally by choosing the label static, which presents the bias and targets input, but stochastic procedure's strength is not severely established for unbiased statics exclusive based targets information. The proposed values are performed on the input [0, 1] by applying the stochastic SCJGNNs procedure for the numerical performances of the MDMPM. The neuron structure using the input, output and hidden layers is shown in Figure 4.

    Figure 4.  Different layers structure for the MDMPM.

    The numerical solution for three different cases of the MDMPM by applying the computing procedures based on the SCJGNNs is presented in this section.

    Case 1: Consider the values ψ=0.038, αh=0.027, γ1=0.020, ξ=0.13, β1=1730, γ2=0.010, δ1=0.001, ηh=0.0004, δ2=0.001, p=0.611, β2=0.001, ε=0.05, υ=0.01, e=0.022, αm=0.13, γ3=0.072, Sh(0)=0.01, Eh(0)=0.02, Ih(0)=0.03, Rh(0)=0.04, Sm(0)=0.05, and Im(0)=0.06 in system (1), which is shown as:

    {Sh(x)=0.027+0.0014Rh(x)(0.00076Ih(x)+0.0014+0.0013Im(x))Sh(x),(Sh)0=0.01,Eh(x)=(0.00076Ih(x)+0.0013Im(x))Sh(x)0.0024Eh(x),(Eh)0=0.02,Ih(x)=0.00752Ih(x)+0.001Eh(x),(Ih)0=0.03,Rh(x)=0.00611Ih(x)0.00177Rh(x)+0.001Eh(x)+0.001Sh(x),(Rh)0=0.04,Sm(x)=0.130.05Sm(x)0.001584Ih(x)Sm(x),(Sm)0=0.05,Im(x)=0.001584Sm(x)Ih(x)0.05Im(x),(Im)0=0.06. (2)

    Case 2: Consider the parameter values ψ=0.038, αh=0.027, γ1=0.020, ξ=0.13, β1=1730, γ2=0.010, δ1=0.001, ηh=0.0004, δ2=0.001, p=0.611, β2=0.001, ε=0.05, υ=0.01, e=0.022, αm=0.13, γ3=0.072, Sh(0)=0.02, Eh(0)=0.03, Ih(0)=0.04, Rh(0)=0.05, Sm(0)=0.06, and Im(0)=0.07 in system (1) that is given as:

    {Sh(x)=0.027+0.0014Rh(x)(0.00076Ih(x)+0.0014+0.0013Im(x))Sh(x),(Sh)0=0.02,Eh(x)=(0.00076Ih(x)+0.0013Im(x))Sh(x)0.0024Eh(x),(Eh)0=0.03,Ih(x)=0.00752Ih(x)+0.001Eh(x),(Ih)0=0.04,Rh(x)=0.00611Ih(x)0.00177Rh(x)+0.001Eh(x)+0.001Sh(x),(Rh)0=0.05,Sm(x)=0.130.05Sm(x)0.001584Ih(x)Sm(x),(Sm)0=0.06,Im(x)=0.001584Sm(x)Ih(x)0.05Im(x),(Im)0=0.07. (3)

    Case 3: Consider ψ=0.038, αh=0.027, γ1=0.020, ξ=0.13, β1=1730, γ2=0.010, δ1=0.001, ηh=0.0004, δ2=0.001, p=0.611, β2=0.001, ε=0.05, υ=0.01, e=0.022, αm=0.13, γ3=0.072, Sh(0)=0.03, Eh(0)=0.04, Ih(0)=0.05, Rh(0)=0.06, Sm(0)=0.07, and Im(0)=0.08 in system (1), given as:

    {Sh(x)=0.027+0.0014Rh(x)(0.00076Ih(x)+0.0014+0.0013Im(x))Sh(x),(Sh)0=0.03,Eh(x)=(0.00076Ih(x)+0.0013Im(x))Sh(x)0.0024Eh(x),(Eh)0=0.04,Ih(x)=0.00752Ih(x)+0.001Eh(x),(Ih)0=0.05,Rh(x)=0.00611Ih(x)0.00177Rh(x)+0.001Eh(x)+0.001Sh(x),(Rh)0=0.06,Sm(x)=0.130.05Sm(x)0.001584Ih(x)Sm(x),(Sm)0=0.07,Im(x)=0.001584Sm(x)Ih(x)0.05Im(x),(Im)0=0.08. (4)

    The plots of the estimated results based on the numerical SCJGNNs procedure for solving the MDMPM are shown in Figures 5 to 9. Figure 5 presents the best validation measures and state transitions for the nonlinear dynamics of the MDMPM through the stochastic computing SCJGNNs scheme. The values of the mean square error (MSE) and the state of transition values are depicted in Figure 5. The computed performances of MSE using the best curves plots, certification, and training are provided in Figure 5(a-c) and the SoT measures are portrayed in Figure 5(d-f). The best solution performances of the MDMPM are shown at epochs 100, 86 and 69, which are found around 1.69774 ×10−07, 1.4324 × 10−07 and 3.5876 × 10−07. It indicates that convergence through the MSE is observed based on dataset of training, testing and certification. It is also observed that by updating the epochs, the authentications, training, and testing graphs lead to steady-state position up to 10-07. The gradient curves by applying the process of SCJGNNs to solve the MDMPM are calculated as 1.4161×10-06, 8.1804 × 10−07 and 2.9486 × 10−06. These graphic curves indicate the precision, exactness, and convergence by using the SCJGNNs procedure for the MDMPM. An error gradient presents the direction and magnitude that is calculated by the designed neural network for updating the weights in the proper amount/direction. In neural network fitting, the process of backpropagation is used to compute the loss function gradient. Mu is the used for the network's training for the momentum parameter, which includes the updated weight expressions to avoid the local minimum issues. It is observed that the network can stuck sometimes to the local minimum, which does not perform the convergence. In addition, Mu shows the control parameter based on that scheme that is used in the neural network's training.

    Figure 5.  Best validation measures and state transitions for the nonlinear dynamics of the MDMPM by applying the stochastic computing SCJGNNs procedure.
    Figure 6.  Function fitness values and error histogram performances to solve the MDMPM by applying the SCJGNNs.
    Figure 7.  Values of regression for the MDMPM (1).
    Figure 8.  Values of regression for the MDMPM (2).
    Figure 9.  Values of regression for the MDMPM (3).

    Figure 6(a to c) presents the curves of fitting for the nonlinear MDMPM, which certifies the comparison of the results. Figure 6(d-f) signifies the error histogram (EH) values for the MDMPM by applying the computing SCJGNNs. The construction of EH is used to test the errors of the targeted and predicted values after the designed artificial neural network training. These errors present that how predicted measures contrast from the targeted values. These values based on the EH are obtained as 1.95 × 10−05, 8.64 × 10−05 and −5.90 × 10−06 for respective dynamics of the MDMPM. Figures 7 to 9 present the correlation values for the MDMPM using the SCJGNNs. These measures based on the determination coefficient R2 are found as 1 for each case of the MDMPM. The graphs of substantiation, testing, and training verify the correctness of the MDMPM using the numerical stochastic SCJGNNs procedure. The values of MSE convergence using the train, epochs, certification, backpropagation, test, and intricacy are tabulated in Table 2. The complexity measures using the SCJGNNs for the MDMPM are performed to authenticate the intricacy based on the network's training, i.e., epoch execution.

    Table 2.  MSE performances for solving the MDMPM.
    Case MSE for instances of sampling Gradient Performance Epoch Time
    Test validation Train
    I 2.745×10−07 1.697 ×10−07 2.366×10−07 1.42×10−06 2.26×10−07 100 2 Sec
    II 1.795×10−07 1.432×10−07 1.920×10−07 8.18×10−07 1.91×10−07 86 2 Sec
    III 5.112×10−07 3.038×10−07 3.678×10−07 2.94×10−06 2.09×10−07 69 2 Sec

     | Show Table
    DownLoad: CSV

    Figures 10 and 11 present the comparison graphs and AE using the obtained and source statics for the MDMPM. It is demonstrated that the source and calculated outcomes of the MDMPM precisely coincided. Figure 10 represents the AE graphs for each category of the MDMPM. For the first class, the AE lies as 10−03 to 10−05 for case I and III and 10−02 to 10−04 for case II. These values for the 2nd category of the MDMPM are found around 10−05 to 10−06, 10−04 to 10−06 and 10−03 to 10−05 for cases I, II and III. The AE measures for the 3rd dynamics of the MDMPM lie as 10−04 to 10−07, 10−03 to 10−05 and 10−03 to 10−06 for respective cases. These values for the 4th class of the MDMPM are calculated as 10−03 to 10−05 for case I and for case II and III, these AE are found as 10−03 to 10−04. The AE values for the 5th and 6th dynamics are found as 10−03 to 10−05 for each case of the model. These reduceable AE measures provides the accuracy of the scheme for the MDMPM.

    Figure 10.  Achieved and reference results of the nonlinear MDMPM.
    Figure 11.  The values of the AE for the nonlinear MDMPM.

    The stochastic numerical performances of the malaria disease using the pesticides and medication have been presented in this study. The populations-based host and vector are divided in the mathematical classification of the malaria using the pesticides and medication. Some conclusions of this work are reported as:

    • The stochastic scale conjugate gradient neural network framework has been designed successfully for the numerical malaria disease model.

    • The neural network procedure is presented by using fifteen neurons along with the log-sigmoid transfer function in the hidden layers to solve the model.

    • The supervised neural networks process using the data as for validation 12%, testing 10%, and training 78% is presented for the numerical solutions of the system.

    • The accuracy of the proposed stochastic scheme is examined through the overlapping of the achieved obtained and Runge-Kutta source solutions.

    • The negligible performances of AE indicate the exactness of the computing stochastic approach.

    • The consistency and trustworthiness of the designed procedure are assessed through the correlation, histogram curves, regression, and functional fitness process to solve the MDMPM.

    • The perks/advantages of the proposed stochastic SCJGNNs procedure includes simplicity of the concept and understanding, straightforwardness in implementations and execution, provision of outcome on continuous domain, expandability and applicability.

    The limitation/disadvantage of the stochastic SCJGNNs paradigm is the issue of trapping in local minima for scale conjugate gradient methodology while training of the weights of the networks, however, the said limitation is rectified by exploited heuristic global search methodologies in future by the interested readers. Additionally, in future, the designed SCJGNNs procedure can be explored/implemented for different applications of the nonlinear mathematical models arising in the field of computational fluid mechanics.

    The authors declare that they have no conflict of interest

    Manal Alqhtani: Conceptualization, methodology, validation, Writing-original draft preparation, writing-review; J.F. Gómez-Aguilar: Conceptualization, methodology, validation, investigation, Writing-original draft preparation, writing-review; Khaled M. Saad: Conceptualization, methodology, validation, writing-review and editing; Zulqurnain Sabir: Validation, investigation, Writing-original draft preparation, writing-review; Eduardo Pérez-Careta: Validation, formal analysis, investigation. All authors have read and agreed to the published version of the manuscript.

    The authors are thankful to the Deanship of Scientific Research at Najran University for funding this work under the General Research Funding program grant code (NU/RG/SERC/12/22).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.



    [1] M. O. Albertson, Chromatic number, independence ratio, and crossing number, ARS Math. Contemp., 1 (2008), 1–6. https://doi.org/10.26493/1855-3974.10.2d0 doi: 10.26493/1855-3974.10.2d0
    [2] C. Bachmaier, F. J. Brandenburg, K. Hanauer, D. Neuwirth, J. Reislhuber, NIC-planar graphs, Discrete Appl. Math., 232 (2017), 23–40. https://doi.org/10.1016/j.dam.2017.08.015 doi: 10.1016/j.dam.2017.08.015
    [3] J. A. Bondy, U. S. R. Murty, Graph theory, GTM 244, Springer, 2008.
    [4] J. Barát, G. Tóth, Improvements on the density of maximal 1-planar graphs, J. Graph Theory, 88 (2018), 101–109. https://doi.org/10.1002/jgt.22187 doi: 10.1002/jgt.22187
    [5] S. N. Bhatt, F. T. Leighton, A framework for solving VLSI graph layout problems, J. Comput. Syst. Sci., 28 (1984), 300–343. https://doi.org/10.1016/0022-0000(84)90071-0 doi: 10.1016/0022-0000(84)90071-0
    [6] G. L. Chia, K. A. Sim, On the skewness of the join of graphs, Discrete Appl. Math., 161 (2013), 2405–2409. https://doi.org/10.1016/j.dam.2013.03.023 doi: 10.1016/j.dam.2013.03.023
    [7] M. Chimani, C. Gutwenger, Non-planar core reduction of graphs, Discrete Math., 309 (2009), 1838–1855. https://doi.org/10.1016/j.disc.2007.12.078 doi: 10.1016/j.disc.2007.12.078
    [8] R. J. Cimikowski, Graph planarization and skewness, Congr. Numer., 88 (1992), 21–32.
    [9] S. Cabello, B. Mohar, Adding one edge to planar graphs makes crossing number and 1-planarity hard, SIAM J. Comput., 42 (2013), 1803–1829. https://doi.org/10.1137/120872310 doi: 10.1137/120872310
    [10] Z. Ding, Z. Ouyang, Y. Huang, F. M. Dong, New upper bounds for the crossing numbers of crossing-critical graphs, Discrete Math., 345 (2022), 113090. https://doi.org/10.1016/j.disc.2022.113090 doi: 10.1016/j.disc.2022.113090
    [11] M. R. Garey, D. S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods, 4 (1983), 312–316. https://doi.org/10.1137/0604033 doi: 10.1137/0604033
    [12] A. Liebers, Planarizing graphs–A survey and annotated bibliography, J. Graph Algorithms Appl., 5 (2001), 1–74.
    [13] P. C. Liu, R. C. Geldmacher, On the deletion of nonplanar edges of a graph, Congressus Numerantium, 24 (1979), 727–738.
    [14] D. McQuillan, S. Pan, R. B. Richter, On the crossing number of K13, J. Combin. Theory, Ser. B, 115 (2015), 224–235. https://doi.org/10.1016/j.jctb.2015.06.002 doi: 10.1016/j.jctb.2015.06.002
    [15] M. Schaefer, Crossing numbers of graphs, CRC Press, Florida, 2017.
    [16] X. Zhang, G. Liu, The structure of plane graphs with independent crossings and its application to coloring problems, Centr. Eur. J. Math., 11 (2013), 308–321. https://doi.org/10.2478/s11533-012-0094-7 doi: 10.2478/s11533-012-0094-7
  • This article has been cited by:

    1. Ahmed Alsaedi, Bashir Ahmad, Afrah Assolami, Sotiris K. Ntouyas, On a nonlinear coupled system of differential equations involving Hilfer fractional derivative and Riemann-Liouville mixed operators with nonlocal integro-multi-point boundary conditions, 2022, 7, 2473-6988, 12718, 10.3934/math.2022704
    2. M. Vellappandi, Venkatesan Govindaraj, José Vanterler da C. Sousa, Fractional optimal reachability problems with ψ ‐Hilfer fractional derivative , 2022, 45, 0170-4214, 6255, 10.1002/mma.8168
    3. Fidel Meléndez-Vázquez, Guillermo Fernández-Anaya, Aldo Jonathan Muñóz-Vázquez, Eduardo Gamaliel Hernández-Martínez, Generalized conformable operators: Application to the design of nonlinear observers, 2021, 6, 2473-6988, 12952, 10.3934/math.2021749
    4. Hamid Lmou, Khalid Hilal, Ahmed Kajouni, Serkan Araci, A New Result for ψ -Hilfer Fractional Pantograph-Type Langevin Equation and Inclusions, 2022, 2022, 2314-4785, 1, 10.1155/2022/2441628
    5. Ravi P. Agarwal, Afrah Assolami, Ahmed Alsaedi, Bashir Ahmad, Existence Results and Ulam–Hyers Stability for a Fully Coupled System of Nonlinear Sequential Hilfer Fractional Differential Equations and Integro-Multistrip-Multipoint Boundary Conditions, 2022, 21, 1575-5460, 10.1007/s12346-022-00650-6
    6. Abdulwasea Alkhazzan, Wadhah Al-Sadi, Varaporn Wattanakejorn, Hasib Khan, Thanin Sitthiwirattham, Sina Etemad, Shahram Rezapour, A new study on the existence and stability to a system of coupled higher-order nonlinear BVP of hybrid FDEs under the p-Laplacian operator, 2022, 7, 2473-6988, 14187, 10.3934/math.2022782
    7. Sina Etemad, Iram Iqbal, Mohammad Esmael Samei, Shahram Rezapour, Jehad Alzabut, Weerawat Sudsutad, Izzet Goksel, Some inequalities on multi-functions for applying in the fractional Caputo–Hadamard jerk inclusion system, 2022, 2022, 1029-242X, 10.1186/s13660-022-02819-8
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1537) PDF downloads(88) Cited by(0)

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog