Theoretical Computer Science and Discrete Mathematics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 35721

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Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43003 Tarragona, Spain
Interests: graph theory; applied mathematics; discrete mathematics
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Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
Interests: graph theory; applied mathematics; discrete mathematics; computer science
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Special Issue Information

This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim is to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The issue covers topics in discrete mathematics including (but not limited to) graph theory, coding theory, cryptography, algorithms and complexity, discrete optimization, discrete geometry, and computational geometry. Contributions presented to the issue can be original research papers, short notes or surveys.

Prof. Dr. Juan Alberto Rodríguez Velázquez
Dr. Alejandro Estrada-Moreno
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Keywords

  • graph theory
  • coding theory
  • cryptography
  • algorithms and complexity
  • discrete optimization
  • discrete geometry and computational geometry

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Published Papers (16 papers)

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26 pages, 1661 KiB  
Article
Modeling and Optimization for Multi-Objective Nonidentical Parallel Machining Line Scheduling with a Jumping Process Operation Constraint
by Guangyan Xu, Zailin Guan, Lei Yue, Jabir Mumtaz and Jun Liang
Symmetry 2021, 13(8), 1521; https://doi.org/10.3390/sym13081521 - 18 Aug 2021
Cited by 2 | Viewed by 2107
Abstract
This paper investigates the nonidentical parallel production line scheduling problem derived from an axle housing machining workshop of an axle manufacturer. The characteristics of axle housing machining lines are analyzed, and a nonidentical parallel line scheduling model with a jumping process operation (NPPLS-JP), [...] Read more.
This paper investigates the nonidentical parallel production line scheduling problem derived from an axle housing machining workshop of an axle manufacturer. The characteristics of axle housing machining lines are analyzed, and a nonidentical parallel line scheduling model with a jumping process operation (NPPLS-JP), which considers mixed model production, machine eligibility constraints, and fuzzy due dates, is established so as to minimize the makespan and earliness/tardiness penalty cost. While the physical structures of the parallel lines in the NPPLS-JP model are symmetric, the production capacities and process capabilities are asymmetric for different models. Different from the general parallel line scheduling problem, NPPLS-JP allows for a job to transfer to another production line to complete the subsequent operations (i.e., jumping process operations), and the transfer is unidirectional. The significance of the NPPLS-JP model is that it meets the demands of multivariety mixed model production and makes full use of the capacities of parallel production lines. Aiming to solve the NPPLS-JP problem, we propose a hybrid algorithm named the multi-objective grey wolf optimizer based on decomposition (MOGWO/D). This new algorithm combines the GWO with the multi-objective evolutionary algorithm based on decomposition (MOEA/D) to balance the exploration and exploitation abilities of the original MOEA/D. Furthermore, coding and decoding rules are developed according to the features of the NPPLS-JP problem. To evaluate the effectiveness of the proposed MOGWO/D algorithm, a set of instances with different job scales, job types, and production scenarios is designed, and the results are compared with those of three other famous multi-objective optimization algorithms. The experimental results show that the proposed MOGWO/D algorithm exhibits superiority in most instances. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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20 pages, 359 KiB  
Article
An Improved Crow Search Algorithm Applied to the Phase Swapping Problem in Asymmetric Distribution Systems
by Brandon Cortés-Caicedo, Laura Sofía Avellaneda-Gómez, Oscar Danilo Montoya, Lázaro Alvarado-Barrios and César Álvarez-Arroyo
Symmetry 2021, 13(8), 1329; https://doi.org/10.3390/sym13081329 - 23 Jul 2021
Cited by 10 | Viewed by 2083
Abstract
This paper discusses the power loss minimization problem in asymmetric distribution systems (ADS) based on phase swapping. This problem is presented using a mixed-integer nonlinear programming model, which is resolved by applying a master–slave methodology. The master stage consists of an improved version [...] Read more.
This paper discusses the power loss minimization problem in asymmetric distribution systems (ADS) based on phase swapping. This problem is presented using a mixed-integer nonlinear programming model, which is resolved by applying a master–slave methodology. The master stage consists of an improved version of the crow search algorithm. This stage is based on the generation of candidate solutions using a normal Gaussian probability distribution. The master stage is responsible for providing the connection settings for the system loads using integer coding. The slave stage uses a power flow for ADSs based on the three-phase version of the iterative sweep method, which is used to determine the network power losses for each load connection supplied by the master stage. Numerical results on the 8-, 25-, and 37-node test systems show the efficiency of the proposed approach when compared to the classical version of the crow search algorithm, the Chu and Beasley genetic algorithm, and the vortex search algorithm. All simulations were obtained using MATLAB and validated in the DigSILENT power system analysis software. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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10 pages, 270 KiB  
Article
From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs
by Ana Almerich-Chulia, Abel Cabrera Martínez, Frank Angel Hernández Mira and Pedro Martin-Concepcion
Symmetry 2021, 13(7), 1282; https://doi.org/10.3390/sym13071282 - 16 Jul 2021
Cited by 1 | Viewed by 1703
Abstract
Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of vV(G). Let f:V(G){0,1,2} be [...] Read more.
Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of vV(G). Let f:V(G){0,1,2} be a function and Vi={vV(G):f(v)=i} for every i{0,1,2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V1V2 has no isolated vertex and N(v)V2 for every vV(G)\V2. The strongly total Roman domination number of G, denoted by γtRs(G), is defined as the minimum weight ω(f)=xV(G)f(x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing γtRs(G) is NP-hard. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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15 pages, 374 KiB  
Article
Optimal Demand Reconfiguration in Three-Phase Distribution Grids Using an MI-Convex Model
by Oscar Danilo Montoya, Andres Arias-Londoño, Luis Fernando Grisales-Noreña, José Ángel Barrios and Harold R. Chamorro
Symmetry 2021, 13(7), 1124; https://doi.org/10.3390/sym13071124 - 24 Jun 2021
Cited by 12 | Viewed by 1887
Abstract
The problem of the optimal load redistribution in electrical three-phase medium-voltage grids is addressed in this research from the point of view of mixed-integer convex optimization. The mathematical formulation of the load redistribution problem is developed in terminals of the distribution node by [...] Read more.
The problem of the optimal load redistribution in electrical three-phase medium-voltage grids is addressed in this research from the point of view of mixed-integer convex optimization. The mathematical formulation of the load redistribution problem is developed in terminals of the distribution node by accumulating all active and reactive power loads per phase. These loads are used to propose an objective function in terms of minimization of the average unbalanced (asymmetry) grade of the network with respect to the ideal mean consumption per-phase. The objective function is defined as the l1-norm which is a convex function. As the constraints consider the binary nature of the decision variable, each node is conformed by a 3×3 matrix where each row and column have to sum 1, and two equations associated with the load redistribution at each phase for each of the network nodes. Numerical results demonstrate the efficiency of the proposed mixed-integer convex model to equilibrate the power consumption per phase in regards with the ideal value in three different test feeders, which are composed of 4, 15, and 37 buses, respectively. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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17 pages, 516 KiB  
Article
Quasi-Ordinarization Transform of a Numerical Semigroup
by Maria Bras-Amorós, Hebert Pérez-Rosés and José Miguel Serradilla-Merinero
Symmetry 2021, 13(6), 1084; https://doi.org/10.3390/sym13061084 - 17 Jun 2021
Viewed by 1699
Abstract
In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, [...] Read more.
In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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14 pages, 297 KiB  
Article
From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs
by Abel Cabrera Martínez, Alejandro Estrada-Moreno and Juan Alberto Rodríguez-Velázquez
Symmetry 2021, 13(6), 1036; https://doi.org/10.3390/sym13061036 - 8 Jun 2021
Cited by 3 | Viewed by 1692
Abstract
This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex [...] Read more.
This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex xV(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set XV(G) is defined to be N(X)=xXN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)X. Now, for every set XV(G) and every vertex xX, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={yV(G)X:N(y)X={x}}. Let Xw={xX:Pe(x,X)}. The strong differential of X is defined to be s(X)=|Ne(X)||Xw|, while the quasi-total strong differential of G is defined to be s*(G)=max{s(X):XV(G)andXwN(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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27 pages, 4577 KiB  
Article
Improving Multivariate Microaggregation through Hamiltonian Paths and Optimal Univariate Microaggregation
by Armando Maya-López, Fran Casino and Agusti Solanas
Symmetry 2021, 13(6), 916; https://doi.org/10.3390/sym13060916 - 21 May 2021
Cited by 5 | Viewed by 2157
Abstract
The collection of personal data is exponentially growing and, as a result, individual privacy is endangered accordingly. With the aim to lessen privacy risks whilst maintaining high degrees of data utility, a variety of techniques have been proposed, being microaggregation a very popular [...] Read more.
The collection of personal data is exponentially growing and, as a result, individual privacy is endangered accordingly. With the aim to lessen privacy risks whilst maintaining high degrees of data utility, a variety of techniques have been proposed, being microaggregation a very popular one. Microaggregation is a family of perturbation methods, in which its principle is to aggregate personal data records (i.e., microdata) in groups so as to preserve privacy through k-anonymity. The multivariate microaggregation problem is known to be NP-Hard; however, its univariate version could be optimally solved in polynomial time using the Hansen-Mukherjee (HM) algorithm. In this article, we propose a heuristic solution to the multivariate microaggregation problem inspired by the Traveling Salesman Problem (TSP) and the optimal univariate microaggregation solution. Given a multivariate dataset, first, we apply a TSP-tour construction heuristic to generate a Hamiltonian path through all dataset records. Next, we use the order provided by this Hamiltonian path (i.e., a given permutation of the records) as input to the Hansen-Mukherjee algorithm, virtually transforming it into a multivariate microaggregation solver we call Multivariate Hansen-Mukherjee (MHM). Our intuition is that good solutions to the TSP would yield Hamiltonian paths allowing the Hansen-Mukherjee algorithm to find good solutions to the multivariate microaggregation problem. We have tested our method with well-known benchmark datasets. Moreover, with the aim to show the usefulness of our approach to protecting location privacy, we have tested our solution with real-life trajectories datasets, too. We have compared the results of our algorithm with those of the best performing solutions, and we show that our proposal reduces the information loss resulting from the microaggregation. Overall, results suggest that transforming the multivariate microaggregation problem into its univariate counterpart by ordering microdata records with a proper Hamiltonian path and applying an optimal univariate solution leads to a reduction of the perturbation error whilst keeping the same privacy guarantees. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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12 pages, 289 KiB  
Article
Study of Parameters in the Genetic Algorithm for the Attack on Block Ciphers
by Osmani Tito-Corrioso, Miguel Angel Borges-Trenard, Mijail Borges-Quintana, Omar Rojas and Guillermo Sosa-Gómez
Symmetry 2021, 13(5), 806; https://doi.org/10.3390/sym13050806 - 5 May 2021
Cited by 3 | Viewed by 2387
Abstract
In recent years, the use of Genetic Algorithms (GAs) in symmetric cryptography, in particular in the cryptanalysis of block ciphers, has increased. In this work, the study of certain parameters that intervene in GAs was carried out, such as the time it takes [...] Read more.
In recent years, the use of Genetic Algorithms (GAs) in symmetric cryptography, in particular in the cryptanalysis of block ciphers, has increased. In this work, the study of certain parameters that intervene in GAs was carried out, such as the time it takes to execute a certain number of iterations, so that a number of generations to be carried out in an available time can be estimated. Accordingly, the size of the set of individuals that constitute admissible solutions for GAs can be chosen. On the other hand, several fitness functions were introduced, and which ones led to better results was analyzed. The experiments were performed with the block ciphers AES(t), for t{3,4,7}. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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22 pages, 806 KiB  
Article
Determination of a Good Indicator for Estimated Prime Factor and Its Modification in Fermat’s Factoring Algorithm
by Rasyid Redha Mohd Tahir, Muhammad Asyraf Asbullah, Muhammad Rezal Kamel Ariffin and Zahari Mahad
Symmetry 2021, 13(5), 735; https://doi.org/10.3390/sym13050735 - 21 Apr 2021
Cited by 6 | Viewed by 1953
Abstract
Fermat’s Factoring Algorithm (FFA) is an integer factorisation methods factoring the modulus N using exhaustive search. The appearance of the Estimated Prime Factor (EPF) method reduces the cost of FFA’s loop count. However, the EPF does not work for balanced primes. This paper [...] Read more.
Fermat’s Factoring Algorithm (FFA) is an integer factorisation methods factoring the modulus N using exhaustive search. The appearance of the Estimated Prime Factor (EPF) method reduces the cost of FFA’s loop count. However, the EPF does not work for balanced primes. This paper proposed the modified Fermat’s Factoring Algorithm 1-Estimated Prime Factor (mFFA1-EPF) that improves the EPF method. The algorithm works for factoring a modulus with unbalanced and balanced primes, respectively. The main results of mFFA1-EPF focused on three criteria: (i) the approach to select good candidates from a list of convergent continued fraction, (ii) the establishment of new potential initial values based on EPF, and (iii) the application of the above modification upon FFA. The resulting study shows the significant improvement that reduces the loop count of FFA1 via (improved) EPF compared to existing methods. The proposed algorithm can be executed without failure and caters for both the modulus N with unbalanced and balanced primes factor. The algorithm works for factoring a modulus with unbalanced and balanced primes. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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25 pages, 2854 KiB  
Article
Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam
by Martin Sotola, Pavel Marsalek, David Rybansky, Martin Fusek and Dusan Gabriel
Symmetry 2021, 13(4), 712; https://doi.org/10.3390/sym13040712 - 18 Apr 2021
Cited by 16 | Viewed by 3906
Abstract
Topology optimization is a modern method for optimizing the material distribution in a given space, automatically searching for the ideal design of the product. The method aims to maximize the design performance of the system regarding given conditions. In engineering practice, a given [...] Read more.
Topology optimization is a modern method for optimizing the material distribution in a given space, automatically searching for the ideal design of the product. The method aims to maximize the design performance of the system regarding given conditions. In engineering practice, a given space is first described using the finite element method and, subsequently, density-based method with solid isotropic material with penalty. Then, the final shape is found using a gradient-based method, such as the optimality criteria algorithm. However, obtaining the ideal shape is highly dependent on the correct setting of numerical parameters. This paper focuses on the sensitivity analysis of key formulations of topology optimization using the implementation of mathematical programming techniques in MATLAB software. For the purposes of the study, sensitivity analysis of a simple spatial task—cantilever bending—is performed. This paper aims to present the formulations of the optimization problem—in this case, minimization of compliance. It should be noted that this paper does not present any new mathematical formulas but rather provides an introduction into the mathematical theory (including filtering methods and calculating large-size problems using the symmetry of matrices) as well as a step-by step guideline for the minimization of compliance within the density-based topology optimization and search for an optimal shape. The results can be used for complex commercial applications produced by traditional manufacturing processes or by additive manufacturing methods. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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10 pages, 265 KiB  
Article
Cryptanalysis of a Group Key Establishment Protocol
by Jorge Martínez Carracedo and Adriana Suárez Corona
Symmetry 2021, 13(2), 332; https://doi.org/10.3390/sym13020332 - 17 Feb 2021
Viewed by 1814
Abstract
In this paper, we analyze the security of a group key establishment scheme proposed by López-Ramos et al. This proposal aims at allowing a group of users to agree on a common key. We present several attacks against the security of the proposed [...] Read more.
In this paper, we analyze the security of a group key establishment scheme proposed by López-Ramos et al. This proposal aims at allowing a group of users to agree on a common key. We present several attacks against the security of the proposed protocol. In particular, an active attack is presented, and it is also proved that the protocol does not provide forward secrecy. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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11 pages, 302 KiB  
Article
Secure w-Domination in Graphs
by Abel Cabrera Martínez, Alejandro Estrada-Moreno and Juan A. Rodríguez-Velázquez
Symmetry 2020, 12(12), 1948; https://doi.org/10.3390/sym12121948 - 25 Nov 2020
Cited by 4 | Viewed by 1997
Abstract
This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where [...] Read more.
This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where w=(w0,w1,,wl) is a vector of nonnegative integers such that w01. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of vV(G). We say that a function f:V(G){0,1,,l} is a w-dominating function if f(N(v))=uN(v)f(u)wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=vV(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,uV(G) with f(v)=0 and f(u)>0, the function fuv is defined by fuv(v)=1, fuv(u)=f(u)1 and fuv(x)=f(x) for every xV(G)\{u,v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f(v)=0, there exists uN(v) such that f(u)>0 and fuv is a w-dominating function as well. The secure w-domination number of G, denoted by γws(G), is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on γws(G) and raises the challenge of conducting a detailed study of the topic. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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11 pages, 284 KiB  
Article
Total Domination in Rooted Product Graphs
by Abel Cabrera Martínez and Juan A. Rodríguez-Velázquez
Symmetry 2020, 12(11), 1929; https://doi.org/10.3390/sym12111929 - 23 Nov 2020
Cited by 7 | Viewed by 2412
Abstract
During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 [...] Read more.
During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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12 pages, 269 KiB  
Article
On the Outer-Independent Roman Domination in Graphs
by Abel Cabrera Martínez, Suitberto Cabrera García, Andrés Carrión García and Angela María Grisales del Rio
Symmetry 2020, 12(11), 1846; https://doi.org/10.3390/sym12111846 - 9 Nov 2020
Cited by 6 | Viewed by 1909
Abstract
Let G be a graph with no isolated vertex and f:V(G){0,1,2} a function. Let [...] Read more.
Let G be a graph with no isolated vertex and f:V(G){0,1,2} a function. Let Vi={vV(G):f(v)=i} for every i{0,1,2}. The function f is an outer-independent Roman dominating function on G if V0 is an independent set and every vertex in V0 is adjacent to at least one vertex in V2. The minimum weight ω(f)=vV(G)f(v) among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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9 pages, 261 KiB  
Article
The n-Pythagorean Fuzzy Sets
by Anna Bryniarska
Symmetry 2020, 12(11), 1772; https://doi.org/10.3390/sym12111772 - 26 Oct 2020
Cited by 10 | Viewed by 2297
Abstract
The following paper presents deductive theories of n-Pythagorean fuzzy sets (n-PFS). N-PFS objects are a generalization of the intuitionistic fuzzy sets (IFSs) and the Yager Pythagorean fuzzy sets (PFSs). Until now, the values of membership and non-membership functions have been described on a [...] Read more.
The following paper presents deductive theories of n-Pythagorean fuzzy sets (n-PFS). N-PFS objects are a generalization of the intuitionistic fuzzy sets (IFSs) and the Yager Pythagorean fuzzy sets (PFSs). Until now, the values of membership and non-membership functions have been described on a one-to-one scale and a quadratic function scale. There is a symmetry between the values of this membership and non-membership functions. The scales of any power functions are used here in order to increase the scope of the decision-making problems. The theory of n-PFS introduces a conceptual apparatus analogous to the classic theory of Zadeh fuzzy sets, consistently striving to correctly define the n-PFS algebra. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)

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3 pages, 190 KiB  
Correction
Correction: Cabrera Martínez et al. On the Secure Total Domination Number of Graphs. Symmetry 2019, 11, 1165
by Abel Cabrera Martínez, Luis P. Montejano and Juan A. Rodríguez-Velázquez
Symmetry 2021, 13(9), 1668; https://doi.org/10.3390/sym13091668 - 10 Sep 2021
Viewed by 1205
Abstract
The authors wish to make the following corrections on paper [...] Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
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