American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
A dominating set is a split dominating
set in . If the induced subgraph is
disconnected in The split domination number of
is denoted by , is the minimum cardinality of
a split dominating set in . In this paper, some results on
were obtained in terms of vertices, blocks, and other
different parameters of but not members of
Further, we develop its relationship with other different
domination parameters of
The document discusses the Pythagorean theorem and its applications. It covers the theorem itself, which states that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. It also covers the converse, special right triangles, using the theorem to solve problems and in coordinate geometry, and extending the theorem to circles.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
This document provides information about sequences and series. It defines sequences as functions with positive integers as the domain. It distinguishes between infinite and finite sequences. Examples of sequences are provided and explicit formulas for finding terms are derived. Methods for finding the nth term of arithmetic and geometric sequences are described.
This document discusses representing functions as power series and manipulating power series. It begins by introducing power series representations of functions like 1/(1-x) and provides examples of finding power series for other functions. It then covers differentiating and integrating power series term-by-term, which preserves the radius of convergence. Examples demonstrate finding power series for derivatives and integrals of other functions. The document emphasizes that power series representations are only valid within their radius of convergence.
This document provides a lesson on drawing polygons on the coordinate plane. It includes 4 examples of plotting points and drawing polygons to connect those points. It then provides the name of each polygon drawn and how to calculate its area, whether by using formulas for basic shapes like triangles or decomposing complex shapes into simpler ones. The document emphasizes using coordinates to determine side lengths and plotting points accurately on the coordinate plane in order to find polygon areas.
A dominating set is a split dominating
set in . If the induced subgraph is
disconnected in The split domination number of
is denoted by , is the minimum cardinality of
a split dominating set in . In this paper, some results on
were obtained in terms of vertices, blocks, and other
different parameters of but not members of
Further, we develop its relationship with other different
domination parameters of
The document discusses the Pythagorean theorem and its applications. It covers the theorem itself, which states that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. It also covers the converse, special right triangles, using the theorem to solve problems and in coordinate geometry, and extending the theorem to circles.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
This document provides information about sequences and series. It defines sequences as functions with positive integers as the domain. It distinguishes between infinite and finite sequences. Examples of sequences are provided and explicit formulas for finding terms are derived. Methods for finding the nth term of arithmetic and geometric sequences are described.
This document discusses representing functions as power series and manipulating power series. It begins by introducing power series representations of functions like 1/(1-x) and provides examples of finding power series for other functions. It then covers differentiating and integrating power series term-by-term, which preserves the radius of convergence. Examples demonstrate finding power series for derivatives and integrals of other functions. The document emphasizes that power series representations are only valid within their radius of convergence.
This document provides a lesson on drawing polygons on the coordinate plane. It includes 4 examples of plotting points and drawing polygons to connect those points. It then provides the name of each polygon drawn and how to calculate its area, whether by using formulas for basic shapes like triangles or decomposing complex shapes into simpler ones. The document emphasizes using coordinates to determine side lengths and plotting points accurately on the coordinate plane in order to find polygon areas.
This document discusses Karnaugh maps (K-maps), a method for simplifying Boolean algebra expressions. It begins by stating that K-maps allow for minimized results with less calculation compared to Boolean algebra alone. The document then covers the basics of K-maps, including how to represent different numbers of variables and the rules for grouping ones and zeros. It provides examples of using K-maps to minimize functions with 2 to 5 variables. Finally, it discusses extensions of K-maps, such as incorporating don't cares, using maxterms instead of minterms, and when the Quine-McCluskey method is preferable to K-maps for problems with many variables.
The document discusses calculating the probability (P) that the first number in a set of numbers is the smallest. It discretizes the intervals into small units of length ε. Pk is defined as the probability the first number is smallest if it is in the kth ε-unit. Approximations are made to express Pk as e-k^2/2. P is then expressed as the sum of these Pk terms, which is approximated as an integral and solved to be π/2√N, where N is the total number of intervals.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document contains solutions to 5 math problems proved using mathematical induction and other techniques:
1) It proves via induction that the sum of the given fractions equals n/(n+1) for all n in the natural numbers.
2) It proves via induction that 5n-1 is divisible by 4 for all n in the natural numbers.
3) It proves that if sets A and B have sizes n and m respectively, with m>n, then no function from A to B can be onto.
4) It proves that the function f(a,b) = (b,a) mapping A x B to B x A is both 1-1 and onto, and therefore the
- Kruskal's algorithm finds a minimum spanning tree by greedily adding edges to a forest in order of increasing weight, as long as it does not form a cycle.
- It runs in O(m log m + n) time by sorting edges first and then using efficient data structures to test for cycles in constant time per edge.
- Prim's algorithm grows a minimum spanning tree from a single vertex by always adding the lowest weight edge that connects a new vertex. It runs in O(n^2) time with basic implementations but can be optimized.
In this note we will see how representations of the two-dimensional Unitary Group U(2) used long ago by the Author for the Many Electron Problem, gives rise to an equilateral triangle in a torus that also defines a Theta Function with Equiharmonic frequencies and whose vertices are up and down spins leading to spin-orbit coupling
This document outlines the axioms and properties of the natural number system as developed by Giuseppe Peano. It defines key terms like successor, addition, multiplication, subtraction, and division on the natural numbers. The five postulates for the natural numbers are: 1 is a natural number; every natural number has a successor; 1 is not a successor; if two numbers have the same successor they are equal; and the principle of mathematical induction. Axioms for the closure, commutativity, and associativity of addition and multiplication are also provided.
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
An Extension to the Zero-Inflated Generalized Power Series Distributionsinventionjournals
In many sampling involving non negative integer data models, the number of zeros is observed to be significantly higher than the expected number in the assumed model. Such models are called zero-inflated models. These models are recently cited in literature in various fields of science including; engineering, natural, social and political sciences. The class of zero-inflated power series distributions was recently considered and studied. Members of the class of generalized power series distributions are some of the well-known discrete distributions such as; the Poisson, binomial, negative binomials, as well as most of their modified forms. In this paper an extension to class of zero-inflated power series distributions was introduced, namely, the zero-one inflated case, and its structural properties were studied. In particular, its moments and some of its moment’s recurrence relations were obtained along with some of its generating functions. Some of these results were shown in term of the zero-inflated cases as well
This document discusses how to simplify powers of monomials. It provides two rules:
1) To raise a power to a power, multiply the exponents and keep the same base. For example, (bm)n = bmn.
2) To find the power of a product, raise each factor to that power. For example, abm= ambm.
It also notes that expressions like (2x)3 and 2x3 are not the same - the first involves multiplying 2x by itself 3 times while the second involves raising 2 to the power of 3 and x to the power of 3 separately. Students are directed to practice problems on page 205.
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
This document contains notes on graphing trigonometric functions. It reviews the domains, ranges, periods, maximums, minimums, and intercepts of y=sin(x), y=cos(x), and y=tan(x). It explains that compressing or stretching the period of these functions is achieved by including a coefficient b in front of x, such as y=sin(bx). Examples are provided and homework assignments are listed at the end.
The document discusses modeling relationships between quantities using functions. It provides examples of expressing relations as tables and graphs and defining the domain and range. It explains that a function assigns each input to exactly one output and the graph of a function consists of ordered pairs of the input and corresponding output. The document is intended to help students understand how to model relationships between variables.
This document introduces and investigates the concept of contra-#rg-continuous functions between topological spaces. It defines contra-#rg-continuity and related concepts like contra-#rg-irresolute functions. Several properties of contra-#rg-continuous functions are proven, including that every contra-continuous function is contra-#rg-continuous, and the composition of a contra-#rg-continuous function with a continuous function is contra-#rg-continuous. Examples are provided to show certain concepts like contra-#rg-continuity and #rg-continuity are independent. The relationship between contra-#rg-continuity and other types of generalized continuous functions is also examined.
This document presents a fixed point theorem for weakly biased mappings in fuzzy metric spaces. It begins with introductions and definitions related to concepts like weakly compatible maps, weakly biased maps, and the E.A. property. It then proves a lemma showing that if two pairs of self-maps satisfy certain conditions, including one pair having the E.A. property and one range being closed, then the pairs have a coincidence point. The main result is a theorem stating that if two pairs of self-maps are weakly biased in the specified directions and one range is closed, then the maps have a unique common fixed point.
Prim's algorithm is a greedy algorithm that finds a minimum spanning tree in a weighted graph. It works by iteratively adding the lowest weight edge that connects an isolated vertex to the growing spanning tree. The algorithm starts with a single vertex and adds the lowest cost edge that extends the tree until all vertices are connected. The document provides an example of applying Prim's algorithm to find the minimum spanning tree of a weighted graph.
Markov numbers are positive integers that satisfy the Markov Diophantine equation: x2 + y2 + z2 = 3xyz. Only a few Markov numbers exist below 100, and they have been extensively studied. Key results include:
- Markov numbers satisfy several number theoretic properties like being mutually co-prime.
- The distribution of Markov numbers below a given value x has been bounded.
- The Markov and Euclid trees describe the relationships between Markov triples.
- Several theorems prove properties like uniqueness of Markov numbers under certain conditions.
Algorithm Performance For Chessboard Separation ProblemsTina Gabel
The document discusses algorithms for solving variations of chessboard placement problems, specifically the N + k Queens Problem. It presents results comparing the performance of sequential and parallel programs that use backtracking and dancing links algorithms to count solutions. Dancing links was found to outperform backtracking for larger problem sizes. The parallel dancing links implementation distributed work by assigning each processor a unique starting position for the first pawn.
Chessboard Puzzles Part 2 - IndependenceDan Freeman
This document is part 2 of a 4-part series on the mathematics of chessboard puzzles. It examines the concept of independence among different chess pieces on a chessboard, which is defined as the maximum number of pieces that can be placed without any attacking each other. Formulas for the independence number are derived for rooks, bishops, kings, and knights on both square and rectangular boards. The 8-queens problem and its generalization to n-queens are also discussed. Proofs are provided for many of the independence numbers and permutation counts.
This document summarizes a lecture on satisfiability and NP-completeness. It introduces the satisfiability problem and shows that 3-SAT, the problem of determining if a Boolean formula consisting of clauses with 3 literals can be satisfied, is NP-complete by reducing SAT to it. It then shows other problems like vertex cover and maximum clique are NP-complete by reducing 3-SAT to them. Reductions preserve the computational difficulty of problems.
The document discusses connected dominating sets and short cycles. It begins by explaining that excluding longer cycles makes related problems easier to solve. Specifically, it shows that on graphs with girth at least five, high degree vertices must be in any minimum dominating set. However, this does not hold for connected dominating sets, since connectivity must also be maintained. It then describes how to obtain fixed-parameter tractable algorithms for connected dominating set problems by guessing the minimum dominating set and extending it. It also shows that these problems do not admit polynomial kernels by providing a reduction from Fair Connected Colors, which is W-hard.
This document discusses Karnaugh maps (K-maps), a method for simplifying Boolean algebra expressions. It begins by stating that K-maps allow for minimized results with less calculation compared to Boolean algebra alone. The document then covers the basics of K-maps, including how to represent different numbers of variables and the rules for grouping ones and zeros. It provides examples of using K-maps to minimize functions with 2 to 5 variables. Finally, it discusses extensions of K-maps, such as incorporating don't cares, using maxterms instead of minterms, and when the Quine-McCluskey method is preferable to K-maps for problems with many variables.
The document discusses calculating the probability (P) that the first number in a set of numbers is the smallest. It discretizes the intervals into small units of length ε. Pk is defined as the probability the first number is smallest if it is in the kth ε-unit. Approximations are made to express Pk as e-k^2/2. P is then expressed as the sum of these Pk terms, which is approximated as an integral and solved to be π/2√N, where N is the total number of intervals.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document contains solutions to 5 math problems proved using mathematical induction and other techniques:
1) It proves via induction that the sum of the given fractions equals n/(n+1) for all n in the natural numbers.
2) It proves via induction that 5n-1 is divisible by 4 for all n in the natural numbers.
3) It proves that if sets A and B have sizes n and m respectively, with m>n, then no function from A to B can be onto.
4) It proves that the function f(a,b) = (b,a) mapping A x B to B x A is both 1-1 and onto, and therefore the
- Kruskal's algorithm finds a minimum spanning tree by greedily adding edges to a forest in order of increasing weight, as long as it does not form a cycle.
- It runs in O(m log m + n) time by sorting edges first and then using efficient data structures to test for cycles in constant time per edge.
- Prim's algorithm grows a minimum spanning tree from a single vertex by always adding the lowest weight edge that connects a new vertex. It runs in O(n^2) time with basic implementations but can be optimized.
In this note we will see how representations of the two-dimensional Unitary Group U(2) used long ago by the Author for the Many Electron Problem, gives rise to an equilateral triangle in a torus that also defines a Theta Function with Equiharmonic frequencies and whose vertices are up and down spins leading to spin-orbit coupling
This document outlines the axioms and properties of the natural number system as developed by Giuseppe Peano. It defines key terms like successor, addition, multiplication, subtraction, and division on the natural numbers. The five postulates for the natural numbers are: 1 is a natural number; every natural number has a successor; 1 is not a successor; if two numbers have the same successor they are equal; and the principle of mathematical induction. Axioms for the closure, commutativity, and associativity of addition and multiplication are also provided.
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
An Extension to the Zero-Inflated Generalized Power Series Distributionsinventionjournals
In many sampling involving non negative integer data models, the number of zeros is observed to be significantly higher than the expected number in the assumed model. Such models are called zero-inflated models. These models are recently cited in literature in various fields of science including; engineering, natural, social and political sciences. The class of zero-inflated power series distributions was recently considered and studied. Members of the class of generalized power series distributions are some of the well-known discrete distributions such as; the Poisson, binomial, negative binomials, as well as most of their modified forms. In this paper an extension to class of zero-inflated power series distributions was introduced, namely, the zero-one inflated case, and its structural properties were studied. In particular, its moments and some of its moment’s recurrence relations were obtained along with some of its generating functions. Some of these results were shown in term of the zero-inflated cases as well
This document discusses how to simplify powers of monomials. It provides two rules:
1) To raise a power to a power, multiply the exponents and keep the same base. For example, (bm)n = bmn.
2) To find the power of a product, raise each factor to that power. For example, abm= ambm.
It also notes that expressions like (2x)3 and 2x3 are not the same - the first involves multiplying 2x by itself 3 times while the second involves raising 2 to the power of 3 and x to the power of 3 separately. Students are directed to practice problems on page 205.
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
This document contains notes on graphing trigonometric functions. It reviews the domains, ranges, periods, maximums, minimums, and intercepts of y=sin(x), y=cos(x), and y=tan(x). It explains that compressing or stretching the period of these functions is achieved by including a coefficient b in front of x, such as y=sin(bx). Examples are provided and homework assignments are listed at the end.
The document discusses modeling relationships between quantities using functions. It provides examples of expressing relations as tables and graphs and defining the domain and range. It explains that a function assigns each input to exactly one output and the graph of a function consists of ordered pairs of the input and corresponding output. The document is intended to help students understand how to model relationships between variables.
This document introduces and investigates the concept of contra-#rg-continuous functions between topological spaces. It defines contra-#rg-continuity and related concepts like contra-#rg-irresolute functions. Several properties of contra-#rg-continuous functions are proven, including that every contra-continuous function is contra-#rg-continuous, and the composition of a contra-#rg-continuous function with a continuous function is contra-#rg-continuous. Examples are provided to show certain concepts like contra-#rg-continuity and #rg-continuity are independent. The relationship between contra-#rg-continuity and other types of generalized continuous functions is also examined.
This document presents a fixed point theorem for weakly biased mappings in fuzzy metric spaces. It begins with introductions and definitions related to concepts like weakly compatible maps, weakly biased maps, and the E.A. property. It then proves a lemma showing that if two pairs of self-maps satisfy certain conditions, including one pair having the E.A. property and one range being closed, then the pairs have a coincidence point. The main result is a theorem stating that if two pairs of self-maps are weakly biased in the specified directions and one range is closed, then the maps have a unique common fixed point.
Prim's algorithm is a greedy algorithm that finds a minimum spanning tree in a weighted graph. It works by iteratively adding the lowest weight edge that connects an isolated vertex to the growing spanning tree. The algorithm starts with a single vertex and adds the lowest cost edge that extends the tree until all vertices are connected. The document provides an example of applying Prim's algorithm to find the minimum spanning tree of a weighted graph.
Markov numbers are positive integers that satisfy the Markov Diophantine equation: x2 + y2 + z2 = 3xyz. Only a few Markov numbers exist below 100, and they have been extensively studied. Key results include:
- Markov numbers satisfy several number theoretic properties like being mutually co-prime.
- The distribution of Markov numbers below a given value x has been bounded.
- The Markov and Euclid trees describe the relationships between Markov triples.
- Several theorems prove properties like uniqueness of Markov numbers under certain conditions.
Algorithm Performance For Chessboard Separation ProblemsTina Gabel
The document discusses algorithms for solving variations of chessboard placement problems, specifically the N + k Queens Problem. It presents results comparing the performance of sequential and parallel programs that use backtracking and dancing links algorithms to count solutions. Dancing links was found to outperform backtracking for larger problem sizes. The parallel dancing links implementation distributed work by assigning each processor a unique starting position for the first pawn.
Chessboard Puzzles Part 2 - IndependenceDan Freeman
This document is part 2 of a 4-part series on the mathematics of chessboard puzzles. It examines the concept of independence among different chess pieces on a chessboard, which is defined as the maximum number of pieces that can be placed without any attacking each other. Formulas for the independence number are derived for rooks, bishops, kings, and knights on both square and rectangular boards. The 8-queens problem and its generalization to n-queens are also discussed. Proofs are provided for many of the independence numbers and permutation counts.
This document summarizes a lecture on satisfiability and NP-completeness. It introduces the satisfiability problem and shows that 3-SAT, the problem of determining if a Boolean formula consisting of clauses with 3 literals can be satisfied, is NP-complete by reducing SAT to it. It then shows other problems like vertex cover and maximum clique are NP-complete by reducing 3-SAT to them. Reductions preserve the computational difficulty of problems.
The document discusses connected dominating sets and short cycles. It begins by explaining that excluding longer cycles makes related problems easier to solve. Specifically, it shows that on graphs with girth at least five, high degree vertices must be in any minimum dominating set. However, this does not hold for connected dominating sets, since connectivity must also be maintained. It then describes how to obtain fixed-parameter tractable algorithms for connected dominating set problems by guessing the minimum dominating set and extending it. It also shows that these problems do not admit polynomial kernels by providing a reduction from Fair Connected Colors, which is W-hard.
INDEPENDENT DOMINATION NUMBER OF EULER TOTIENT CAYLEY GRAPHS AND ARITHMETIC G...IAEME Publication
Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated wi th the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph.
N-Queens Combinatorial Problem - Polyglot FP for Fun and Profit – Haskell and...Philip Schwarz
First see the problem solved using the List monad and a Scala for comprehension.
Then see the Scala program translated into Haskell, both using a do expressions and using a List comprehension.
Understand how the Scala for comprehension is desugared, and what role the withFilter function plays.
Also understand how the Haskell do expressions and List comprehension are desugared, and what role the guard function plays.
Scala code for Part 1: https://github.com/philipschwarz/n-queens-combinatorial-problem-scala-part-1
Errata: on slide 30, the resulting lists should be Haskell ones rather than Scala ones.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
1. The document studies the matching domination parameters of Euler Totient Cayley graphs. It defines Euler Totient Cayley graphs and presents theorems about the domination number and matching domination number for various cases like when n is prime, a power of a prime, or neither prime nor a power of a prime.
2. Theorems and proofs are presented for the matching domination number when n is prime, a power of a prime, of the form kp where k is an odd prime, and neither prime nor a power of a prime.
3. Examples illustrating the matching dominating sets for various n are shown through figures.
The document discusses the NP-hard Max Cut problem and provides a reduction from the NP-hard NAE-3-SAT problem to Max Cut to prove that Max Cut is also NP-hard. The reduction works by mapping clauses in a NAE-3-SAT instance to a graph instance of Max Cut, such that a solution to one problem can be translated to a solution for the other problem in polynomial time. This shows that any polynomial time algorithm for Max Cut could also be used to solve NAE-3-SAT in polynomial time. The document then provides a simple randomized approximation algorithm for Max Cut that runs in linear time.
Analysis & Design of Algorithms
Backtracking
N-Queens Problem
Hamiltonian circuit
Graph coloring
A presentation on unit Backtracking from the ADA subject of Engineering.
In this paper, we extend the concept of fair secure dominating sets by characterizing the corona of two nontrivial connected graphs and give some important results.
Introduction to graph theory (All chapter)sobia1122
1) Graph theory can be used to model and solve problems in many fields like physics, chemistry, computer science, and more. Certain problems can be formulated as problems in graph theory.
2) Graph theory has developed from puzzles and practical problems, like the Königsberg bridge problem inspiring Eulerian graph theory and the "Around the World" game inspiring Hamiltonian graph theory.
3) Connectivity in graphs measures how connected a graph is, and how the removal of vertices or edges affects connectivity. Connectivity is important for applications like communication networks.
Kakuro: Solving the Constraint Satisfaction ProblemVarad Meru
This work was done as a part of the project for the course CS 271: Introduction to Artificial Intelligence (http://www.ics.uci.edu/~kkask/Fall-2014%20CS271/index.html), taught in Fall 2014.
The document discusses approximation algorithms for NP-hard optimization problems. It provides examples of approximation algorithms for problems like set cover, vertex cover, traveling salesman problem (TSP), and knapsack. For set cover, it shows that a greedy algorithm provides a (1+ln n)-approximation. For vertex cover and TSP, it describes 2-approximation algorithms. It also presents a fully polynomial-time approximation scheme (FPTAS) for knapsack that provides a solution within (1-eps) of optimal.
Backtracking is an algorithmic-technique for solving problems recursively by trying to build a solution incrementally, one piece at a time, removing those solutions that fail to satisfy the constraints of the problem at any point of time
The document contains 16 multiple choice questions about algorithms, data structures, and graph theory. Each question has 4 possible answers and the correct answer is provided. The maximum number of comparisons needed to merge sorted sequences is 358, and depth first search on a graph represented with an adjacency matrix has a worst case time complexity of O(n^2).
The document discusses the backtracking search algorithm. It begins by introducing backtracking as a method for systematically trying various sequences of decisions until finding one that works. It then describes the backtrack algorithm as a depth-first recursive search that tests for solutions, returns any found, recursively tries choices, and returns failure if no choices remain. An example of the 8 queens problem is provided to demonstrate backtracking by placing queens on a chessboard without any attacking each other. The summary explores pruning search spaces and avoiding hopeless paths to find solutions more efficiently.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
I am Craig D. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Stochastic Processes.
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Stability criterion of periodic oscillations in a (13)Alexander Decker
This document discusses domination problems on isosceles triangular chessboards using different chess pieces. It examines placing a minimum number of pieces such that all unoccupied positions are attacked (the domination number). For a single piece type, it determines the domination number and possible solutions for rooks, bishops, and kings on isosceles triangular boards. It also considers domination numbers when using two piece types together, such as kings and rooks or kings and bishops. Key results include formulas for the domination number and total solutions in terms of the board size for each piece type.
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
The document summarizes a textbook called "Rural Property Valuation" published by the Appraisal Institute. It discusses how the textbook provides essential guidance for appraising rural properties, covering topics unique to valuing agricultural land, ranches, timberland, and other rural uses. The textbook is said to be an important resource for students and practitioners to understand rural markets and issues. It contains 19 chapters covering key appraisal topics from a rural perspective, and provides up-to-date information on valuation of rural properties in the United States.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
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G374044.pdf
1. American Journal of Multidisciplinary Research & Development (AJMRD)
Volume 03, Issue 07 (July- 2021), PP 40-44
ISSN: 2360-821X
www.ajmrd.com
Multidisciplinary Journal www.ajmrd.com Page | 40
Research Paper Open Access
Anti-domination Number of a Graph
MallikarjunBasannaKattimani
Department of Mathematics, The Oxford College of Engineering,
Bangalore-560 068, Karnataka State-INDIA,
Corresponding Author: MallikarjunBasannaKattimani
Abstract: Let be a graph, a set is a of , if every vertex in
is adjacent to at least one vertex in . The of is the minimum
cardinality of a dominating set [1] and [6].
We define, Let be a minimum dominating set of . A set of vertices in of is an
with respect to , if every vertex in is adjacent to at least one vertex in . The
of is the minimum cardinality of an is well
defined. In this paper we obtained exact values of of for some standard graphs and also we
establish some general results and Nordhus and Guddamm type result on this new parameter.
Key words: Dominating set, Anti-dominating set.
Subject Classification: 05C69
I. Introduction
The Graph considered here are finite, connected, nontrivial, undirected, without loops or multiple
edges. Any undefined term in this paper may be found in Harary [4].
A vertex in a graph is said to be dominate every vertex adjacent to it. A set of
vertices in is a , if every vertex in is dominated by at least one vertex in .
Dominating sets were defined by Berge [1] (Where they are called externally stable sets) and Ore [6].
The of a graph is the smallest number of vertices in any minimal
dominating set. It appears in various puzzle questions. In a regular chessboard and the five chess pieces: Rook,
Bishop. Knight, King and Queen all these must tour the board using only legal moves, landing on every square
exactly once. One instance is the so called Five queens problem on the chessboard: It is required to place five
queens on the board in such positions that they dominate each square as shown in (fig.1), no smaller number of
queens will suffice, so that . In 1850’s five is the minimum number of queens that can dominate all
of the squares of chessboard. The five queen’s Problem is to find a dominating set of five queens, [1] and
[6].
2. Anti-domination Number of a Graph
Multidisciplinary Journal www.ajmrd.com Page | 41
(a) (b)
Figure 1.
Among the many solutions to this problem, the two in Fig.1 are particularly interesting. In the first
solution (Fig.1a) no queen is dominated by any other queen, while in the second solution (fig.1b) the opposite is
essentially true, every queen is dominated by at least one other queen. The second solution suggests the
following definition: A set of vertices in is a , if every vertex in is
dominated by at least one vertex in . The were first defined and studied by
Cockayne, Dawes and Hedetniemi [3].
Connecting to the above five-queens problem, five queens are placed in such places in a chessboard as
shown in fig.1 that all remaining 59 squares are attacked or occupied by a queen. Hence, every square is
dominated by at least one of the five queens. A set of five queens is called a .
But in fig.1 the queen is placed in such a way that it poses threat for all 59 squares, hence it can rightly
be termed as , so the problem arises in front of us is How to safeguard the 59 squares? by
five queens. Contrary to this in fig.1 the rook is placed in particular places out of 59 squares before the queen
placed, in such a way that Rook poses threat to queen, hence it can rightly be termed as
, and denoted by .
Then, our solution is, before occupying five particular places by a queen (fig.1), we place a smallest
possible number than five in different particular places out of 59 squares by rooks, which poses threat to the
queens. Having this done we can protect all 59 squares as shown in (fig.2a) or (fig.3a) with the initial
arrangement of rooks in a particular position (fig.2a) or (fig.3a). Hence we say that every queen is dominated by
at least one of the three rooks in (fig.2a) or (fig.3a). We call it as over the chessboard.
Similarly, we can use bishop (fig.2b) and (Fig.3b) or knight in (fig. 2c) and (Fig. 3c) and knight instead
of rook as mentioned in the following
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
3. Anti-domination Number of a Graph
Multidisciplinary Journal www.ajmrd.com Page | 42
5-Q-queen 5-Q-queen 5-Q-queen
3-R-rook 3-B-bishop 3-K-knight
(a) (b) (c)
Figure 2
5-Q-queen 5-Q-queen 5-Q-queen
3-R-rook 3-B-bishop 4-K-knight
(a) (b) (c)
Figure 3
In our discussion, three is the minimum number of rooks that can dominate all of the five queens of
chessboard. The three-rook problem is to find an . Hence it motivates.
We define, let be minimum of . A set of vertices in of is an
with respect to , if every vertex in is adjacent to at least one vertex in . The
of is the minimum cardinality of an .
Results
Exact values of for some standard graph are given in Theorem 1.
R Q
Q
Q R
Q
R Q
Q B
Q
Q
B Q
Q B
Q
Q K K
Q
K Q
Q
Q R
Q R
Q R
Q
Q
Q
B
Q B
Q
Q
B
Q
Q
K Q
Q
K Q K
K
Q
4. Anti-domination Number of a Graph
Multidisciplinary Journal www.ajmrd.com Page | 43
Theorem 1.
(i) (1)
(ii) (2)
(iii) (3)
(iv) (4)
(v) (5)
(vi) (6)
Where, is the greatest positive integer not greater than
Theorem 2. For any graph , (7)
Proof. (7) follows from the definition of and
Theorem 3. For any graph , (8)
Proof. Since and from (7), we have . Hence the result.
Theorem 4. For any graph , (9)
Proof. We know that any graph and also therefore and from (7), we
get Hence the result.
The following results are strait forward.
Theorem 5. If is a or or , then
Theorem 6. For any graph , and is a pendent vertex, then , where
is a cut vertex.
Theorem 7. If is connected and then
Proof. Since, we know that [2] and from (9), we get
5. Anti-domination Number of a Graph
Multidisciplinary Journal www.ajmrd.com Page | 44
Theorem 8. Let be a tree such that every cut vertex is adjacent to at least two end vertices. Then
.
Proof. Since, every cut vertex is belongs to and is an end vertex, therefore
Hence the result.
Theorem 9. Let be a graph with vertices, edges and maximum degree , then
.
Proof. We know that , [2]. And from (7) upper bound holds true. Clearly is connected, we
have and . Hence the result.
Theorem 10. Let be a complete bipartite graph. Then
Proof. Let be a complete bipartite graph on vertex sets and such that and . Let D
be a minimum dominating set in . Suppose . Then is adjacent with at least vertices of . And
is adjacent with at least vertices of . Thus Hence Then is dominated by
one vertex of and is dominated by one vertex of . Thus Hence the result.
Reference
[1] C. Berge; Graph and Hypergraphs, North Holland, Amsterdam, 1973.
[2] E. J. Cockayne and S. T. Hedetniemi; Towards a theory of domination in graphs, , 7, 247-
261 (1977).
[3] E. J.Cockayne, R. M. Dawes and S. T. Hedetniemi; Total domination in graphs ,10, 211-
215 (1980).
[4] F. Harary; Addison-Wesley, Reading, Mass. (1969).
[5] J. W.Glaisher, On the problem of Eight Queens, The Phil. Mag., Series 4, 457.[216].
[6] O. Ore; Theory of Graphs, Am Soc. Colloq. Publ., 38, Providence, R1, 1962.