This document provides a constructive characterization of trees with equal total edge domination and double edge domination numbers. It begins with definitions of total edge domination number and double edge domination number. It then proves several observations and results about these numbers in trees. The main result is Theorem 3.1, which proves that for any tree T, the double edge domination number of T is greater than or equal to the total edge domination number of T. It then introduces six types of operations to construct trees with equal numbers. Lemmas 3.2 through 3.4 prove that applying these operations to trees that already have equal numbers results in another tree with equal numbers.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
Abstract: An edge dominating set D of a fuzzy graph G= (σ, µ) is a non-split edge dominating set if the induced fuzzy sub graph H= (<e-d>, σ¢, µ¢) is connected. The split edge domination number γ¢ns(G)or γ¢ns is the minimum fuzzy cardinality of a non-split edge dominating set. In this paper we study a non-split edge dominating set of fuzzy graphs and investigate the relationship of γ¢ns(G)with other known parameter of G. Keywords: Fuzzy graphs, fuzzy domination, fuzzy edge domination, fuzzy non split edge domination number.
Title: Non Split Edge Domination in Fuzzy Graphs
Author: C.Y. Ponnappan, S. Basheer Ahamed, P. Surulinathan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
The document defines several concepts related to domination in fuzzy graphs, including inverse split domination, non-split domination, and their bounds. It presents definitions for terms like dominating set, split dominating set, inverse dominating set, and establishes bounds on various domination numbers. Several theorems are provided with proofs, including that the inverse split domination number of a fuzzy tree is equal to its domination number, and that the inverse non-split domination number of a fuzzy graph is at most its minimum degree minus one. Coedge split and non-split domination in fuzzy graphs are also defined and properties presented.
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
Dynamic Programming Over Graphs of Bounded TreewidthASPAK2014
This document discusses treewidth and algorithms for graphs with bounded treewidth. It contains three parts:
1) Algorithms for bounded treewidth graphs, including showing weighted max independent set and c-coloring can be solved in FPT time parameterized by treewidth using dynamic programming on tree decompositions.
2) Graph-theoretic properties of treewidth such as definitions of treewidth and tree decompositions.
3) Applications of these algorithms to problems like Hamiltonian cycle on general graphs.
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
This document summarizes the concept of bidimensionality and how it can be used to design subexponential algorithms for graph problems on planar and other graph classes. It discusses how bidimensionality can be defined for parameters that are closed under minors or contractions by relating their behavior on grid graphs. It presents examples like vertex cover and dominating set that are bidimensional. It also discusses how bidimensionality can be extended to bounded genus graphs and H-minor free graphs using grid-minor/contraction theorems.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
Abstract: An edge dominating set D of a fuzzy graph G= (σ, µ) is a non-split edge dominating set if the induced fuzzy sub graph H= (<e-d>, σ¢, µ¢) is connected. The split edge domination number γ¢ns(G)or γ¢ns is the minimum fuzzy cardinality of a non-split edge dominating set. In this paper we study a non-split edge dominating set of fuzzy graphs and investigate the relationship of γ¢ns(G)with other known parameter of G. Keywords: Fuzzy graphs, fuzzy domination, fuzzy edge domination, fuzzy non split edge domination number.
Title: Non Split Edge Domination in Fuzzy Graphs
Author: C.Y. Ponnappan, S. Basheer Ahamed, P. Surulinathan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
The document defines several concepts related to domination in fuzzy graphs, including inverse split domination, non-split domination, and their bounds. It presents definitions for terms like dominating set, split dominating set, inverse dominating set, and establishes bounds on various domination numbers. Several theorems are provided with proofs, including that the inverse split domination number of a fuzzy tree is equal to its domination number, and that the inverse non-split domination number of a fuzzy graph is at most its minimum degree minus one. Coedge split and non-split domination in fuzzy graphs are also defined and properties presented.
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
Dynamic Programming Over Graphs of Bounded TreewidthASPAK2014
This document discusses treewidth and algorithms for graphs with bounded treewidth. It contains three parts:
1) Algorithms for bounded treewidth graphs, including showing weighted max independent set and c-coloring can be solved in FPT time parameterized by treewidth using dynamic programming on tree decompositions.
2) Graph-theoretic properties of treewidth such as definitions of treewidth and tree decompositions.
3) Applications of these algorithms to problems like Hamiltonian cycle on general graphs.
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
This document summarizes the concept of bidimensionality and how it can be used to design subexponential algorithms for graph problems on planar and other graph classes. It discusses how bidimensionality can be defined for parameters that are closed under minors or contractions by relating their behavior on grid graphs. It presents examples like vertex cover and dominating set that are bidimensional. It also discusses how bidimensionality can be extended to bounded genus graphs and H-minor free graphs using grid-minor/contraction theorems.
The document discusses applications of graphs with bounded treewidth. It covers the following key points:
1) Courcelle's theorem shows that many NP-complete graph problems can be solved in linear time for graphs of bounded treewidth using monadic second-order logic. This includes problems like independent set, coloring, and Hamiltonian cycle.
2) The treewidth of a graph is closely related to its largest grid minor - graphs with large treewidth contain large grid minors. There are polynomial relationships between treewidth and largest grid minor for planar graphs.
3) Planar graphs have bounded treewidth if and only if they exclude some grid configuration as a contraction. This helps characterize planar graphs of bounded treewidth
The document discusses matroids and parameterized algorithms. It begins with an overview of Kruskal's greedy algorithm for finding a minimum spanning tree in a graph. It then introduces matroids and defines them as structures where greedy algorithms find optimal solutions. Several examples of matroids are provided, including uniform matroids, partition matroids, graphic matroids, and gammoids. The document also presents an alternate definition of matroids using basis families.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document discusses important cuts and separators in graphs and their algorithmic applications. It begins by defining cuts, minimum cuts, and minimal cuts. It then characterizes cuts as edges on the boundary of a set of vertices. The document discusses properties of cuts like submodularity. It introduces the concept of important cuts and separators, proving several properties about them. Importantly, it proves that the number of important cuts of size at most k is at most 4k. The document discusses applications of important cuts and separators to parameterized algorithms.
The document describes an algorithm for finding a (p,q)-partition of a graph G. A (p,q)-partition partitions the vertices of G into clusters such that each cluster has size at most p and cut size at most q. The algorithm works by first showing that finding a (p,q)-partition can be reduced to finding a (p,q)-cluster for each vertex. It then gives a randomized algorithm for finding a (p,q)-cluster for a given vertex v in time 2O(q)nO(1) by reducing the problem to an instance of the satellite problem, which can be solved in polynomial time. The algorithm works by sampling important cuts in the graph to construct an instance of
Distance domination, guarding and covering of maximal outerplanar graphs政謙 陳
This document summarizes research on distance domination, guarding, and covering of maximal outerplanar graphs. It establishes that:
1) The minimum number of guards needed for distance domination (γkd) of triangulations is less than or equal to the minimum number for distance guarding (gkd), which is less than or equal to the minimum number for distance covering (βkd).
2) Every n-vertex maximal outerplanar graph can be guarded by ⌈n/3⌉ guards, which provides a tight bound.
3) Similarly, every n-vertex maximal outerplanar graph can be 2-distance dominated and 2-distance covered by ⌈n/
This document defines and describes concepts related to fuzzy graphs and fuzzy digraphs. Key points include:
- A fuzzy graph is defined by two functions that assign membership values to vertices and edges.
- A fuzzy subgraph has lower or equal membership values for vertices and edges compared to the original graph.
- Effective edges and effective paths only include edges/paths where the membership equals the minimum vertex membership.
- Various graph measures are generalized to fuzzy graphs, such as vertex degree, order, size, and domination number.
- Fuzzy digraphs are defined similarly but with directed edges. Concepts like paths, independence, and domination are extended to fuzzy digraphs.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
The document describes the iterative compression technique for designing fixed-parameter tractable algorithms. It shows how to use a vertex/edge cover, feedback vertex set, or odd cycle transversal of size k+1 as "compression" to design a branching algorithm with running time of O*(ck) for some constant c. This technique involves guessing how an optimal solution of size k interacts with the given size k+1 solution, and branching based on including or excluding vertices in the optimal solution.
The document describes an algorithm for solving the Steiner Tree problem parameterized by treewidth. It uses dynamic programming over a tree decomposition of the input graph to compute the solution. The algorithm builds a dynamic programming table at each node of the tree decomposition. It handles leaf, forget, introduce and join nodes in t^O(t) time by updating/merging partial solutions. This results in an overall running time of t^O(t) to solve the Steiner Tree problem parameterized by treewidth.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
The document discusses algorithms and techniques for analyzing graphs and polynomials. It describes how an adjacency matrix A can be used to count walks and common neighbors in a graph. It also explains how to construct a polynomial that is equal to zero if and only if a graph G contains a path of length k, by creating terms for all walks in G and exploiting the property that a+a=0 in finite fields of characteristic two. This allows evaluating whether the polynomial is zero to determine if G contains the desired path.
This document summarizes a talk on supersymmetric Q-balls and boson stars in (d+1) dimensions. It introduces Q-balls and boson stars as non-topological solitons stabilized by a conserved Noether charge. It discusses properties like existence conditions and the thin wall approximation for Q-balls. For boson stars, it covers different models and properties like rotating and charged boson stars. The document also discusses applications like the AdS/CFT correspondence and holographic superconductors using boson stars in anti-de Sitter spacetime.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
This document provides definitions and propositions related to abstract algebra. It begins by defining a group as a set with a binary operation that is closed, associative, has an identity element, and where each element has an inverse. It then lists several propositions about properties of groups, including that a group has a unique identity and each element has a unique inverse. The document continues defining additional algebraic structures like rings, fields, subgroups, and properties of groups like cyclic groups. It concludes by discussing matrix groups and their properties.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Proportional and decentralized rule mcst gamesvinnief
This document presents two cost allocation rules - the proportional rule and the decentralized rule - for minimum cost spanning extension problems. The proportional rule allocates costs proportionally based on an algorithm that constructs a minimum cost spanning extension while simultaneously allocating link costs. The decentralized rule is based on an algorithm that constructs the extension in a decentralized manner, with links being added one by one and their costs immediately allocated. Both rules are shown to be refinements of the irreducible core defined for these problems. The proportional rule is then characterized axiomatically.
Trees amd properties slide for presentatonSHARANSASI1
The document discusses properties and parameters of graphs and trees. It defines key terms related to trees such as leaf, spanning tree, and properties like every tree having at least two leaves. It also defines graph parameters like diameter, radius, eccentricity, distance, center, and Wiener index. Formulas are given for counting trees and relations between graph parameters like maximum independent set size, matching size, vertex cover size, and edge cover size.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The document discusses applications of graphs with bounded treewidth. It covers the following key points:
1) Courcelle's theorem shows that many NP-complete graph problems can be solved in linear time for graphs of bounded treewidth using monadic second-order logic. This includes problems like independent set, coloring, and Hamiltonian cycle.
2) The treewidth of a graph is closely related to its largest grid minor - graphs with large treewidth contain large grid minors. There are polynomial relationships between treewidth and largest grid minor for planar graphs.
3) Planar graphs have bounded treewidth if and only if they exclude some grid configuration as a contraction. This helps characterize planar graphs of bounded treewidth
The document discusses matroids and parameterized algorithms. It begins with an overview of Kruskal's greedy algorithm for finding a minimum spanning tree in a graph. It then introduces matroids and defines them as structures where greedy algorithms find optimal solutions. Several examples of matroids are provided, including uniform matroids, partition matroids, graphic matroids, and gammoids. The document also presents an alternate definition of matroids using basis families.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document discusses important cuts and separators in graphs and their algorithmic applications. It begins by defining cuts, minimum cuts, and minimal cuts. It then characterizes cuts as edges on the boundary of a set of vertices. The document discusses properties of cuts like submodularity. It introduces the concept of important cuts and separators, proving several properties about them. Importantly, it proves that the number of important cuts of size at most k is at most 4k. The document discusses applications of important cuts and separators to parameterized algorithms.
The document describes an algorithm for finding a (p,q)-partition of a graph G. A (p,q)-partition partitions the vertices of G into clusters such that each cluster has size at most p and cut size at most q. The algorithm works by first showing that finding a (p,q)-partition can be reduced to finding a (p,q)-cluster for each vertex. It then gives a randomized algorithm for finding a (p,q)-cluster for a given vertex v in time 2O(q)nO(1) by reducing the problem to an instance of the satellite problem, which can be solved in polynomial time. The algorithm works by sampling important cuts in the graph to construct an instance of
Distance domination, guarding and covering of maximal outerplanar graphs政謙 陳
This document summarizes research on distance domination, guarding, and covering of maximal outerplanar graphs. It establishes that:
1) The minimum number of guards needed for distance domination (γkd) of triangulations is less than or equal to the minimum number for distance guarding (gkd), which is less than or equal to the minimum number for distance covering (βkd).
2) Every n-vertex maximal outerplanar graph can be guarded by ⌈n/3⌉ guards, which provides a tight bound.
3) Similarly, every n-vertex maximal outerplanar graph can be 2-distance dominated and 2-distance covered by ⌈n/
This document defines and describes concepts related to fuzzy graphs and fuzzy digraphs. Key points include:
- A fuzzy graph is defined by two functions that assign membership values to vertices and edges.
- A fuzzy subgraph has lower or equal membership values for vertices and edges compared to the original graph.
- Effective edges and effective paths only include edges/paths where the membership equals the minimum vertex membership.
- Various graph measures are generalized to fuzzy graphs, such as vertex degree, order, size, and domination number.
- Fuzzy digraphs are defined similarly but with directed edges. Concepts like paths, independence, and domination are extended to fuzzy digraphs.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
The document describes the iterative compression technique for designing fixed-parameter tractable algorithms. It shows how to use a vertex/edge cover, feedback vertex set, or odd cycle transversal of size k+1 as "compression" to design a branching algorithm with running time of O*(ck) for some constant c. This technique involves guessing how an optimal solution of size k interacts with the given size k+1 solution, and branching based on including or excluding vertices in the optimal solution.
The document describes an algorithm for solving the Steiner Tree problem parameterized by treewidth. It uses dynamic programming over a tree decomposition of the input graph to compute the solution. The algorithm builds a dynamic programming table at each node of the tree decomposition. It handles leaf, forget, introduce and join nodes in t^O(t) time by updating/merging partial solutions. This results in an overall running time of t^O(t) to solve the Steiner Tree problem parameterized by treewidth.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
The document discusses algorithms and techniques for analyzing graphs and polynomials. It describes how an adjacency matrix A can be used to count walks and common neighbors in a graph. It also explains how to construct a polynomial that is equal to zero if and only if a graph G contains a path of length k, by creating terms for all walks in G and exploiting the property that a+a=0 in finite fields of characteristic two. This allows evaluating whether the polynomial is zero to determine if G contains the desired path.
This document summarizes a talk on supersymmetric Q-balls and boson stars in (d+1) dimensions. It introduces Q-balls and boson stars as non-topological solitons stabilized by a conserved Noether charge. It discusses properties like existence conditions and the thin wall approximation for Q-balls. For boson stars, it covers different models and properties like rotating and charged boson stars. The document also discusses applications like the AdS/CFT correspondence and holographic superconductors using boson stars in anti-de Sitter spacetime.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
This document provides definitions and propositions related to abstract algebra. It begins by defining a group as a set with a binary operation that is closed, associative, has an identity element, and where each element has an inverse. It then lists several propositions about properties of groups, including that a group has a unique identity and each element has a unique inverse. The document continues defining additional algebraic structures like rings, fields, subgroups, and properties of groups like cyclic groups. It concludes by discussing matrix groups and their properties.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Proportional and decentralized rule mcst gamesvinnief
This document presents two cost allocation rules - the proportional rule and the decentralized rule - for minimum cost spanning extension problems. The proportional rule allocates costs proportionally based on an algorithm that constructs a minimum cost spanning extension while simultaneously allocating link costs. The decentralized rule is based on an algorithm that constructs the extension in a decentralized manner, with links being added one by one and their costs immediately allocated. Both rules are shown to be refinements of the irreducible core defined for these problems. The proportional rule is then characterized axiomatically.
Trees amd properties slide for presentatonSHARANSASI1
The document discusses properties and parameters of graphs and trees. It defines key terms related to trees such as leaf, spanning tree, and properties like every tree having at least two leaves. It also defines graph parameters like diameter, radius, eccentricity, distance, center, and Wiener index. Formulas are given for counting trees and relations between graph parameters like maximum independent set size, matching size, vertex cover size, and edge cover size.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
This document discusses non split locating equitable domination in graphs. It begins with definitions of terms like domination number and non split locating equitable dominating set. It then presents several theorems that establish bounds on the non split locating equitable domination number of a graph based on its properties. These include bounds related to the number of vertices, minimum degree, number of pendant vertices, and whether the graph is regular or a tree. The document also characterizes the graphs that achieve equality in some of the bounds. In general, it analyzes the non split locating equitable domination number and relates it to other graph parameters.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
This document discusses the concept of strong triple connected domination number (stc) of a graph. Some key points:
1. A subset S of vertices is a strong triple connected dominating set if S is a strong dominating set and the induced subgraph <S> is triple connected.
2. The strong triple connected domination number stc(G) is the minimum cardinality of a strong triple connected dominating set.
3. Some standard graphs for which the exact stc value is determined include paths, cycles, complete graphs, wheels, and more.
4. Bounds on stc(G) are established, such as 3 ≤ stc(G) ≤ p-1
Let G be a simple graph with n vertices, and λ1, · · · , λn be the eigenvalues of its adjacent matrix. The Estrada index of G is a graph invariant, defined as EE = i e n i 1 ,, is proposed as a measure of branching in alkanes. In this paper, we obtain two candidates which have the fourth largest EE among trees with n vertices
IRJET- Independent Middle Domination Number in Jump GraphIRJET Journal
This document discusses independent middle domination number (iM(J(G))) in jump graphs. It defines iM(J(G)) as the minimum cardinality of an independent dominating set of the middle graph M(J(G)). The paper obtains several bounds on iM(J(G)) in terms of the vertices, edges, and other parameters of J(G). It also establishes relationships between iM(J(G)) and other domination parameters such as domination number, strong split domination number, and edge domination number. Exact values of iM(J(G)) are determined for some standard jump graphs like paths, cycles, stars, and wheels.
Equi independent equitable domination number of cycle and bistar related graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Bounds on double domination in squares of graphseSAT Journals
Abstract
Let the square of a graphG , denoted by 2 G has same vertex set as in G and every two vertices u and v are joined in 2 G if and
only if they are joined in G by a path of length one or two. A subset D of vertices of 2 G is a double dominating set if every
vertex in 2 G is dominated by at least two vertices of D . The minimum cardinality double dominating set of 2 G is the double
domination number, and is denoted by 2
d G . In this paper, many bounds on 2
d G
were obtained in terms of elements of
G . Also their relationship with other domination parameters were obtained.
Key words: Graph, Square graph, Double dominating set, Double domination number.
Subject Classification Number: AMS-05C69, 05C70.
This document discusses various graph parameters related to distance between vertices in a graph, including diameter, radius, and average distance. It examines how these parameters are defined, how they relate to each other and other graph properties, and how they behave in different graph classes. It also discusses characterizations of graph classes based on distance properties and generalizations to concepts like Steiner trees.
The document summarizes research on accurate and total accurate dominating sets of interval graphs. It begins by introducing interval graphs and defining accurate and total accurate dominating sets. It then presents an algorithm to construct a minimum dominating set of an interval graph. Several theorems are provided about when the constructed dominating set is or is not an accurate dominating set or total accurate dominating set based on properties of the interval family and dominating set. The document focuses on characterizing accurate dominating sets of interval graphs.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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.
The middle graph of a graph G, denoted by
M(G), is a graph whose vertex set is V(G)UE(G) , and two
vertices are adjacent if they are adjacent edges of G or one is a
vertex and other is a edge incident with it . The Line graph of G ,
written L(G), is the simple graph whose vertices are the edges of
G, with ef Є E(L(G)) when e and f have a common end vertex in
G. A set S of vertices of graph M(G) if S is an independent
dominating set of M(G) if S is an independent set and every
vertex not in S is adjacent to a vertex in S. The independent
middle domination number of G, denoted by iM(G) is the
minimum cardinality of an independent dominating set of
M(G).A dominating set D is a connected dominating set if <d> is
connected. The connected domination number, denoted by Ƴc , is
the minimum number of vertices in a connected dominating set.
In this paper many bounds on iL(G) ,iM(G),ƳM(G) were
obtained in terms of element of G, but not in terms of elements of
L(G) or M(G).
The document discusses various topics relating to trees in discrete mathematics. It begins by defining different types of trees such as free trees, rooted trees, and binary trees. It then discusses concepts like the level and height of vertices in rooted trees. Other topics covered include minimum spanning trees, tree traversals, decision trees, isomorphism of trees, and algorithms for testing isomorphism between binary trees.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
Similar to Characterization of trees with equal total edge domination and double edge domination numbers (20)
Abnormalities of hormones and inflammatory cytokines in women affected with p...Alexander Decker
Women with polycystic ovary syndrome (PCOS) have elevated levels of hormones like luteinizing hormone and testosterone, as well as higher levels of insulin and insulin resistance compared to healthy women. They also have increased levels of inflammatory markers like C-reactive protein, interleukin-6, and leptin. This study found these abnormalities in the hormones and inflammatory cytokines of women with PCOS ages 23-40, indicating that hormone imbalances associated with insulin resistance and elevated inflammatory markers may worsen infertility in women with PCOS.
A usability evaluation framework for b2 c e commerce websitesAlexander Decker
This document presents a framework for evaluating the usability of B2C e-commerce websites. It involves user testing methods like usability testing and interviews to identify usability problems in areas like navigation, design, purchasing processes, and customer service. The framework specifies goals for the evaluation, determines which website aspects to evaluate, and identifies target users. It then describes collecting data through user testing and analyzing the results to identify usability problems and suggest improvements.
A universal model for managing the marketing executives in nigerian banksAlexander Decker
This document discusses a study that aimed to synthesize motivation theories into a universal model for managing marketing executives in Nigerian banks. The study was guided by Maslow and McGregor's theories. A sample of 303 marketing executives was used. The results showed that managers will be most effective at motivating marketing executives if they consider individual needs and create challenging but attainable goals. The emerged model suggests managers should provide job satisfaction by tailoring assignments to abilities and monitoring performance with feedback. This addresses confusion faced by Nigerian bank managers in determining effective motivation strategies.
A unique common fixed point theorems in generalized dAlexander Decker
This document presents definitions and properties related to generalized D*-metric spaces and establishes some common fixed point theorems for contractive type mappings in these spaces. It begins by introducing D*-metric spaces and generalized D*-metric spaces, defines concepts like convergence and Cauchy sequences. It presents lemmas showing the uniqueness of limits in these spaces and the equivalence of different definitions of convergence. The goal of the paper is then stated as obtaining a unique common fixed point theorem for generalized D*-metric spaces.
A trends of salmonella and antibiotic resistanceAlexander Decker
This document provides a review of trends in Salmonella and antibiotic resistance. It begins with an introduction to Salmonella as a facultative anaerobe that causes nontyphoidal salmonellosis. The emergence of antimicrobial-resistant Salmonella is then discussed. The document proceeds to cover the historical perspective and classification of Salmonella, definitions of antimicrobials and antibiotic resistance, and mechanisms of antibiotic resistance in Salmonella including modification or destruction of antimicrobial agents, efflux pumps, modification of antibiotic targets, and decreased membrane permeability. Specific resistance mechanisms are discussed for several classes of antimicrobials.
A transformational generative approach towards understanding al-istifhamAlexander Decker
This document discusses a transformational-generative approach to understanding Al-Istifham, which refers to interrogative sentences in Arabic. It begins with an introduction to the origin and development of Arabic grammar. The paper then explains the theoretical framework of transformational-generative grammar that is used. Basic linguistic concepts and terms related to Arabic grammar are defined. The document analyzes how interrogative sentences in Arabic can be derived and transformed via tools from transformational-generative grammar, categorizing Al-Istifham into linguistic and literary questions.
A time series analysis of the determinants of savings in namibiaAlexander Decker
This document summarizes a study on the determinants of savings in Namibia from 1991 to 2012. It reviews previous literature on savings determinants in developing countries. The study uses time series analysis including unit root tests, cointegration, and error correction models to analyze the relationship between savings and variables like income, inflation, population growth, deposit rates, and financial deepening in Namibia. The results found inflation and income have a positive impact on savings, while population growth negatively impacts savings. Deposit rates and financial deepening were found to have no significant impact. The study reinforces previous work and emphasizes the importance of improving income levels to achieve higher savings rates in Namibia.
A therapy for physical and mental fitness of school childrenAlexander Decker
This document summarizes a study on the importance of exercise in maintaining physical and mental fitness for school children. It discusses how physical and mental fitness are developed through participation in regular physical exercises and cannot be achieved solely through classroom learning. The document outlines different types and components of fitness and argues that developing fitness should be a key objective of education systems. It recommends that schools ensure pupils engage in graded physical activities and exercises to support their overall development.
A theory of efficiency for managing the marketing executives in nigerian banksAlexander Decker
This document summarizes a study examining efficiency in managing marketing executives in Nigerian banks. The study was examined through the lenses of Kaizen theory (continuous improvement) and efficiency theory. A survey of 303 marketing executives from Nigerian banks found that management plays a key role in identifying and implementing efficiency improvements. The document recommends adopting a "3H grand strategy" to improve the heads, hearts, and hands of management and marketing executives by enhancing their knowledge, attitudes, and tools.
This document discusses evaluating the link budget for effective 900MHz GSM communication. It describes the basic parameters needed for a high-level link budget calculation, including transmitter power, antenna gains, path loss, and propagation models. Common propagation models for 900MHz that are described include Okumura model for urban areas and Hata model for urban, suburban, and open areas. Rain attenuation is also incorporated using the updated ITU model to improve communication during rainfall.
A synthetic review of contraceptive supplies in punjabAlexander Decker
This document discusses contraceptive use in Punjab, Pakistan. It begins by providing background on the benefits of family planning and contraceptive use for maternal and child health. It then analyzes contraceptive commodity data from Punjab, finding that use is still low despite efforts to improve access. The document concludes by emphasizing the need for strategies to bridge gaps and meet the unmet need for effective and affordable contraceptive methods and supplies in Punjab in order to improve health outcomes.
A synthesis of taylor’s and fayol’s management approaches for managing market...Alexander Decker
1) The document discusses synthesizing Taylor's scientific management approach and Fayol's process management approach to identify an effective way to manage marketing executives in Nigerian banks.
2) It reviews Taylor's emphasis on efficiency and breaking tasks into small parts, and Fayol's focus on developing general management principles.
3) The study administered a survey to 303 marketing executives in Nigerian banks to test if combining elements of Taylor and Fayol's approaches would help manage their performance through clear roles, accountability, and motivation. Statistical analysis supported combining the two approaches.
A survey paper on sequence pattern mining with incrementalAlexander Decker
This document summarizes four algorithms for sequential pattern mining: GSP, ISM, FreeSpan, and PrefixSpan. GSP is an Apriori-based algorithm that incorporates time constraints. ISM extends SPADE to incrementally update patterns after database changes. FreeSpan uses frequent items to recursively project databases and grow subsequences. PrefixSpan also uses projection but claims to not require candidate generation. It recursively projects databases based on short prefix patterns. The document concludes by stating the goal was to find an efficient scheme for extracting sequential patterns from transactional datasets.
A survey on live virtual machine migrations and its techniquesAlexander Decker
This document summarizes several techniques for live virtual machine migration in cloud computing. It discusses works that have proposed affinity-aware migration models to improve resource utilization, energy efficient migration approaches using storage migration and live VM migration, and a dynamic consolidation technique using migration control to avoid unnecessary migrations. The document also summarizes works that have designed methods to minimize migration downtime and network traffic, proposed a resource reservation framework for efficient migration of multiple VMs, and addressed real-time issues in live migration. Finally, it provides a table summarizing the techniques, tools used, and potential future work or gaps identified for each discussed work.
A survey on data mining and analysis in hadoop and mongo dbAlexander Decker
This document discusses data mining of big data using Hadoop and MongoDB. It provides an overview of Hadoop and MongoDB and their uses in big data analysis. Specifically, it proposes using Hadoop for distributed processing and MongoDB for data storage and input. The document reviews several related works that discuss big data analysis using these tools, as well as their capabilities for scalable data storage and mining. It aims to improve computational time and fault tolerance for big data analysis by mining data stored in Hadoop using MongoDB and MapReduce.
1. The document discusses several challenges for integrating media with cloud computing including media content convergence, scalability and expandability, finding appropriate applications, and reliability.
2. Media content convergence challenges include dealing with the heterogeneity of media types, services, networks, devices, and quality of service requirements as well as integrating technologies used by media providers and consumers.
3. Scalability and expandability challenges involve adapting to the increasing volume of media content and being able to support new media formats and outlets over time.
This document surveys trust architectures that leverage provenance in wireless sensor networks. It begins with background on provenance, which refers to the documented history or derivation of data. Provenance can be used to assess trust by providing metadata about how data was processed. The document then discusses challenges for using provenance to establish trust in wireless sensor networks, which have constraints on energy and computation. Finally, it provides background on trust, which is the subjective probability that a node will behave dependably. Trust architectures need to be lightweight to account for the constraints of wireless sensor networks.
This document discusses private equity investments in Kenya. It provides background on private equity and discusses trends in various regions. The objectives of the study discussed are to establish the extent of private equity adoption in Kenya, identify common forms of private equity utilized, and determine typical exit strategies. Private equity can involve venture capital, leveraged buyouts, or mezzanine financing. Exits allow recycling of capital into new opportunities. The document provides context on private equity globally and in developing markets like Africa to frame the goals of the study.
This document discusses a study that analyzes the financial health of the Indian logistics industry from 2005-2012 using Altman's Z-score model. The study finds that the average Z-score for selected logistics firms was in the healthy to very healthy range during the study period. The average Z-score increased from 2006 to 2010 when the Indian economy was hit by the global recession, indicating the overall performance of the Indian logistics industry was good. The document reviews previous literature on measuring financial performance and distress using ratios and Z-scores, and outlines the objectives and methodology used in the current study.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Introducing Milvus Lite: Easy-to-Install, Easy-to-Use vector database for you...Zilliz
Join us to introduce Milvus Lite, a vector database that can run on notebooks and laptops, share the same API with Milvus, and integrate with every popular GenAI framework. This webinar is perfect for developers seeking easy-to-use, well-integrated vector databases for their GenAI apps.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdfMalak Abu Hammad
Discover how MongoDB Atlas and vector search technology can revolutionize your application's search capabilities. This comprehensive presentation covers:
* What is Vector Search?
* Importance and benefits of vector search
* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
* Best practices and optimization strategies
Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
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- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
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UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
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End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
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UiPath Test Automation using UiPath Test Suite series, part 5
Characterization of trees with equal total edge domination and double edge domination numbers
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.5, 2012
Characterization of Trees with Equal Total Edge Domination and
Double Edge Domination Numbers
M.H.Muddebihal A.R.Sedamkar*
Department of Mathematics, Gulbarga University, Gulbarga-585106, Karnataka, INDIA
E-mail of the corresponding author: mhmuddebihal@yahoo.co.in
Abstract
A total edge dominating set of a graph G is a set D of edges of G such that the sub graph D has no isolated
edges. The total edge domination number of G denoted by γ t (G ) , is the minimum cardinality of a total edge
'
dominating set of G . Further, the set D is said to be double edge dominating set of graph G . If every edge of G is
dominated by at least two edges of D . The double edge domination number of G , denoted by, γ d (G ) ,
'
is the
minimum cardinality of a double edge dominating set of G . In this paper, we provide a constructive
characterization of trees with equal total edge domination and double edge domination numbers.
Key words: Trees, Total edge domination number, Double edge domination number.
1. Introduction:
In this paper, we follow the notations of [2]. All the graphs considered here are simple, finite, non-trivial,
undirected and connected graph. As usual p = V and q = E denote the number of vertices and edges of a
graph G , respectively.
In general, we use X to denote the sub graph induced by the set of vertices X and N ( v ) and
N [ v ] denote the open and closed neighborhoods of a vertex v , respectively.
The degree of an edge e = uv of G is defined by deg e = deg u + deg v − 2 , is the number of edges
adjacent to it. An edge e of degree one is called end edge and neighbor is called support edge of G .
A strong support edge is adjacent to at least two end edges. A star is a tree with exactly one vertex of degree
greater than one. A double star is a tree with exactly one support edge.
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For an edge e is a rooted tree T , let C ( e ) and S ( e ) denote the set of childrens and descendants of e
respectively. Further we define S [e] = S (e) U {e} . The maximal sub tree at e is the sub tree of T induced
by S [e] , and is denoted by Te .
A set D ⊆ E is said to be total edge dominating set of G , if the sub graph D has no isolated edges. The
total edge domination number of G , denoted by γ t
'
(G ) , is the minimum cardinality of a total edge dominating set
of G . Total edge domination in graphs was introduced by S.Arumugam and S.Velammal [1].
A set S ⊆ V is said to be double dominating set of G , if every vertex of G is dominated by at least two
vertices of S . The double domination number of G , denoted by γ d (G ) , is the minimum cardinality of a double
dominating set of G . Double domination is a graph was introduced by F. Harary and T. W. Haynes [3]. The concept
of domination parameters is now well studied in graph theory (see [4] and [5]).
Analogously, a set D ⊆ E is said to be double edge dominating set of G , if every edge of G is dominated
by at least two edges of D . The double edge domination number of G , denoted by γ d
'
(G ) , is the minimum
cardinality of a double edge dominating set of G .
In this paper, we provide a constructive characterization of trees with equal total edge domination and
double edge domination numbers.
2. Results:
Initially we obtain the following Observations which are straight forward.
Observation 2.1: Every support edge of a graph G is in every γ t' (G ) set.
Observation 2.2: Suppose every non end edge is adjacent to exactly two end edge, then every end edge of a graph
G is in every γ d (G ) set.
'
Observation 2.3: Suppose the support edges of a graph G are at distance at least three in G , then every support
edge of a graph G is in every γ d (G ) .
'
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3. Main Results:
Theorem 3.1: For any tree T , γ d (T ) ≥ γ t' (T ) .
'
Proof: Let q be the number of edges in tree T . We proceed by induction on q . If diam (T ) ≤ 3 . Then T is either
a star or a double star and γ d
'
(T ) = 2 = γ t' (T ) . Now assume that diam (T ) ≥ 4 and the Theorem is true for
every tree T ' with q ' < q . First assume that some support edge of T , say ex is strong. Let ey and ez be the end
edges adjacent to ex and T ' = T − e . Let D ' be any γ d (T ' ) - set. Clearly ex ∈ D ' , where D ' is a total edge
'
dominating set of tree T . Therefore γ t' (T ' ) ≤ γ t' (T ) . Now let S be any γ d (T )
'
- set. By observations 2 and 3, we
have e y , e x , e z ∈ S . Clearly, S − {ey } is a double edge dominating set of tree T ' . Therefore
γ d (T ' ) ≤ γ d (T ) − 1.
' '
Clearly, γ d (T ) ≥ γ d (T ' ) + 1 ≥ γ t' (T ' ) + 1 ≥ γ t' (T ) + 1 ≥ γ t' (T ) ,
' '
a contradiction.
Therefore every support edge of T is weak.
Let T be a rooted tree at vertex r which is incident with edge er of the diam (T ) . Let et be the end
edge at maximum distance from er , e be parent of et , eu be the parent of e and ew be the parent of eu in the
rooted tree. Let Tex denotes the sub tree induced by an edge ex and its descendents in the rooted tree.
Assume that degT (eu ) ≥ 3 and eu is adjacent to an end edge ex . Let T ' = T − e and D ' be the
γ t' (T ' ) - set. By Observation 1, we have eu ∈ D ' . Clearly, D ' U {e} is a total edge dominating set of tree T . Thus
γ t' (T ) ≤ γ t' (T ' ) + 1 . Now let S be any γ d (T ) -
'
set. By Observations 2 and 3, et , e x , e, eu ∈ S . Clearly,
S − {e, et } is a double edge dominating set of tree T ' . Therefore γ d (T ') ≤ γ d (T ) − 2 .
' '
It follows that
γ d (T ) ≥ γ d (T ' ) + 2 ≥ γ t' (T ' ) + 2 ≥ γ t' (T ) +1 ≥ γ t' (T ) .
' '
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Now assume that among the decedents of eu there is a support edge say ex , which is different from e . Let
T ' = T − e and D ' be the γ t' (T ' ) - set containing no end edges. Since ex must have a neighbor in D ' , thus
eu ∈ D ' . Clearly D ' U {e} is a total edge dominating set of tree T and hence γ t' (T ) ≤ γ t' (T ' ) + 1 . Now let S be
any γ d (T ) - set. By Observations 2 and 3, we have et , e, e x ∈ S . If eu ∈ S , then S − {e, et } is the double edge
'
dominating set of tree T . Further assume that eu ∉ S . Then S U {eu } − {e, et } is a double edge dominating set of
'
tree T . Therefore γ d (T ' ) ≤ γ d (T ) − 1 .
' '
Clearly, it follows that, γ d (T ) ≥ γ d (T ' ) + 1 ≥ γ t' (T ') + 1 ≥ γ t' (T ) .
' '
Therefore, we obtain γ d (T ) ≥ γ t' (T ) .
'
To obtain the characterization, we introduce six types of operations that we use to construct trees with equal total
edge domination and double edge domination numbers.
Type 1: Attach a path P to two vertices u and w which are incident with eu and ew respectively of T where
1
eu , ew located at the component of T − ex ey such that either ex is in γ d set of T or ey is in γ d - set of T .
' '
Type 2: Attach a path P2 to a vertex v incident with e of tree T , where e is an edge such that T − e has a
component P .
3
Type 3: Attach k ≥ 1 number of paths P3 to the vertex v which is incident with an edge e of T where e is an
edge such that either T − e has a component P2 or T − e has two components P2 and P4 , and one end of P4 is
adjacent to e is T .
Type 4: Attach a path P3 to a vertex v which is incident with e of tree T by joining its support vertex to v , such
that e is not contained is any γ t' - set of T .
Type 5: Attach a path P4 ( n ) , n ≥ 1 to a vertex v which is incident with an edge e , where e is in a γ d - set of T
'
if n = 1 .
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Type 6: Attach a path P5 to a vertex v incident with e of tree T by joining one of its support to v such that
T − e has a component H ∈{P3 , P4 , P6 }.
Now we define the following families of trees
Let ℑ be the family of trees with equal total edge domination number and double edge domination number. That is
ℑ= {T / T is a tree satisfying γ t' (T ) = γ d ( T )} .
'
We also define one more family as
ℜ = {T / T is obtained from P3 by a finite sequence of type - i operations where
1 ≤ i ≤ 5} .
We prove the following Lemmas to provide our main characterization.
Lemma 3.2: If T ∈ ℑ and T is obtained from T by Type-1 operation, then T ∈ ℑ .
' '
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ' ) . By Theorem 1, γ t' (T ) ≤ γ d (T ) , we only need to prove that
' '
γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching the path P to two vertices u and w which are
1
incident with eu and ew as eu and ew respectively. Then there is a path ex ey in T ' such that either ex is in γ d -
' ' '
set of T and T ' − ex ey has a component P5 = eu eew ex or ey is in γ d - set of T ' and T ' − ex ey has a
'
component P6 = eu eew ex ex . Clearly,
'
γ t' (T ' ) = γ t' (T ) − 1 .
Suppose T ' − ex ey contains a path P5 = eu eew ex then S ' be the γ d - set of T ' containing ex . From Observation 2
'
and by the definition of γd
'
- set, we have S ' I {eu , e , ew , ex } = {eu , ew } or {eu , e} . Therefore
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S = ( S ' − {eu , e , ew }) U {eu , e, ew } is a double edge dominating set of T with S = S ' + 1 = γ d (T ') + 1 .
' ' '
Clearly, γ t' (T ) = γ t' (T ' ) + 1 = γ d (T ' ) + 1 = S > γ d (T ) .
' '
Now, if T ' − ex ey contains a path P6 = eu e ew ex ex . Then S ' be the γ d - set of T ' containing ey . By Observation 2
' '
and by definition of γd
'
- set, we have S I { eu , e, ew , ex , ex } = { eu , ew , ex } .
' ' '
Therefore
S = ( S ' − {eu , ew }) U {eu , e, ew } is a double edge dominating set of T with S = S ' + 1 = γ d (T ' ) + 1 .
' ' '
Clearly, γ t' (T ) = γ t' (T ') + 1 = γ d (T ' ) +1 = S ≥ γ d (T ) .
' '
Lemma 3.3: If T ' ∈ ℑ and T is obtained from T ' by Type-2 operation, then T ∈ ℑ .
Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t' (T ) ≤ γ d (T ) ,
' ' ' ' ' '
we only need to prove that
γ t' (T ) ≥ γ d (T ) . Assume that T
' '
is obtained from T by attaching a path P2 to a vertex v which is incident with
e of T ' where T ' − e has a component P3 = ewex . We can easily show that γ t' (T ) = γ t' (T ' ) + 1 . Now by
definition of γd
'
- set, there exists a γd
'
- set, D ' of T ' containing the edge e . Then D ' U {eu } forms a double
'
edge dominating set of T . Therefore γ t' (T ) = γ t' (T ' ) + 1 = γ d (T ' ) + 1 = D ' U {eu } ≥ γ d (T ) .
' ' '
Lemma 3.4: If T ∈ ℑ and T is obtained from T by Type - 3 operation, then T ∈ ℑ .
' '
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' '
that γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching m ≥ 1 number of paths P3 to a vertex v
which is incident with an edge e of T ' such that T ' − e has a component P3 or two components P2 and P4 . By
definition of γ t' - set and γd -
'
set, we can easily show that γ t
'
(T ) ≥ γ t' (T ' ) + 2m and γ d (T ') + 2m ≥ γ d (T ) .
' '
Since γ t (T ' ) = γ d (T ') , it follows that γ t' (T ) ≥ γ t' (T ') + 2m = γ d (T ') + 2m ≥ γ d (T ) .
' ' ' '
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Vol.2, No.5, 2012
Lemma 3.5: If T ' ∈ ℑ and T is obtained from T ' by Type - 4 operation, then T ∈ ℑ .
Proof: Since T ∈ ℑ , we have γ t (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' ' ' '
that γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching path P3 to a vertex v incident with e in
T ' such that e is not contained in any γ t' -set of T ' and T '− e has a component P4 . For any γ d - set, S ' of T ' ,
'
S ' U {ex , ez } is a double edge dominating set of T . Hence γ d (T ' ) + 2 ≥ γ d (T ) . Let D be any γ t' - set of
' '
T containing the edge eu , which implies ey ∈ D and D I {e, ex , ez } =1 .
If e ∉ D , then D I E (T ' ) = D − 2 = γ t' (T ) − 2 ≥ γ t' (T ' ) , since D I E (T ' ) is a total edge
T '. since γ t (T ' ) = γ d (T ') , that γ t (T ) ≥ γ t (T ') + 2
' ' ' '
dominating set of Further it follows
= γ d (T ') + 2 ≥ γ d (T ) .
' '
If e∈D , then D I {e, ex , ez } = {e} and D I E (T ' ) = D −1 = γ t' (T ) −1 ≥ γ t' (T ') , since
D I E (T ') is a total edge dominating set of T ' . Suppose γ t' (T ) ≤ γ d (T ) − 1 , then by γ d (T ' ) = γ t' (T ') , it
' '
follows that γ d (T ) ≥ γ t' (T ) + 1≥ γ t' (T ') + 2 = γ d (T ') + 2 ≥ γ d (T ') .
' ' '
Clearly,
D I E (T ') = γ t' (T ) − 1 = γ t' (T ') and D I E (T ') is a total edge dominating set of T '
containing e .
'
Therefore, it gives a contradiction to the fact that e is not in any total edge dominating set of T and
hence γ t (T ) ≥ γ d (T ) .
' '
Lemma 3.6: If T ∈ ℑ and T is obtained from T by Type - 5 operation, then T ∈ ℑ .
' '
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' '
that γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching path P4 ( n ) , n ≥ 1 to a vertex v incident
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with e in T ' such that e is in γ d - set for n = 1 . Clearly, γ t' (T ) ≥ γ t' (T ' ) + 2 n . If n ≥ 2 , then by
'
γ t' (T ' ) = γ d (T ') , it is obvious that γ t' (T ) ≥ γ t' (T ') + 2n = γ d (T ') + 2 n ≥ γ d (T )
' ' '
. If n = 1 , then D ' be a
γd
'
- set of T ' containing e . Thus D ' U {ez , e x } is a double edge dominating set of T . Hence
γ t' (T ) ≥ γ t' (T ') + 2 = γ d (T ') +2 = S ' U {ez , ex } ≥ γ d (T ).
' '
Lemma 3.7: If T ∈ ℑ and T is obtained from T ' by Type - 6 operation, then T ∈ ℑ .
'
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' '
that γ t' (T ) ≥ γ d (T ) .
'
Assume that T is obtained from T ' by attaching path a path P5 to a vertex v which is
incident with e . Then there exists a subset D of E (T ) as γ t' - set of T such that D I NT ' (e) ≠ φ for n = 1 .
Therefore D I E (T ') is a total edge dominating set of T ' and D I E (T ') ≥ γ t' (T ') . By the definition of double
edge dominating set, we have γ d (T ') + 3 ≥ γ d (T )
' '
. Clearly, it follows that
γ t' (T ) = D = D I E ( P6 ) + D I E (T ') > 3 + γ t' (T ') = 3 + γ d (T ') ≥ γ d (T ) .
' '
Now we define one more family as
Let T be the rooted tree. For any edge e ∈ E (T ) , let C ( e ) and F ( e ) denote the set of children edges and
descendent edges of e respectively. Now we define
C ' (e) = {eu ∈ C (e) every edge of F [ eu ] has a distance at most two from e in T } .
Further partition C ' (e) into C1' (e) , C2' (e) and C3' (e) such that every edge of Ci ' (e) has edge
degree i in T , i = 1, 2 and 3 .
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Lemma 3.8: Let T be a rooted tree satisfying γ t' (T ) = γ d (T )
'
and ew ∈ E (T ) . We have the following
conditions:
If C (ew ) ≠ φ , then C1 (ew ) = C3 (ew ) = φ .
' ' '
1.
If C3 (ew ) ≠ φ , then C1' (ew ) = C2' (ew ) = φ and C3' (ew ) = 1 .
'
2.
3. If C (ew ) = C
'
(ew ) ≠ C1' (ew ) , then C1' (ew ) = C3' (ew ) = φ .
Proof: Let C1' (ew ) = {ex1 , ex2 ,...........exl } , C2' (ew ) = {ey1 , ey2 ,...........eym } and C3' (ew ) =
{ez1 , ez2 ,...........ezn } such that C1' (ew ) = l , C2' (ew ) = m and C3' (ew ) = n . For every i = 1, 2,..., n , let
eu be the end edge adjacent to e z in T and T ' = T − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } .
i i
For (1): We prove that if m ≥ 1 , then l + n = 0 . Assume l + n ≥ 1 . Since m ≥ 1 , we can have a γd
'
- set S of T
such that ew ∈ S and a γ t' - set D of
'
T ' such that ew ∈D . Clearly S − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } is a
'
double edge dominating set of T' and D ' is a total edge dominating set of T . Hence
γ t' (T ' ) = D ' ≥ γ t' (T ) = γ d (T ) = S > S − {ex , ex ,..., ex , eu , eu , …, eu } ≥ γ d (T ' ) , it gives a contradiction
'
1 2
'
l 1 2 n
with Theorem 1.
For (2) and (3): Either if C3' (ew ) ≠ φ or if C (ew ) = C ' (ew ) ≠ C1' (ew ) . Then for both cases, m + n ≥ 1 . Now
T ' such that ew ∈ D . Then D ' is also a total edge dominating set of T . Hence
'
select a γ t' - set D of
'
γ t' (T ' ) = D ' ≥ γ t' (T ) . Further by definition of γd
'
- set and by Observation 2, there exists a γd
'
-set S of T
which satisfies S I {ey1 , ey2 ,..., eym , ez1 , ez2 ,..., ezn } = φ . Then ( S I E (T ' )) U {ew } is a γ d -set of T ' . Hence
'
γ d (T ' ) ≤ ( S I E (T ' )) U {ew } ≤ S − (l + n) + 1 = γ d (T ) − (l + n) + 1 = γ t' (T ) − (l + n) + 1 .
' '
If n ≥ 1 , then γ d (T ') ≤ γ t' (T ) ≤ γ t' (T ' ) ≤ γ d (T ' ) , the last inequality is by Theorem 1. It follows that l + n = 1
' '
and ew ∉ S . Therefore l = 0 and n = 1 . From Condition 1, we have m = 0 . Hence 2 follows.
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If C (ew ) = C (ew ) ≠ C1' (ew ) , then m + n ≥ 1 . By conditions 1 and 2, l = 0 . Now we show that n = 0 .
'
Otherwise, similar to the proof of 2, we have ew ∉ S , n = 1 and m = 0 . Since C (ew ) = C ' (ew )
and deg (ew ) = 2 , for double edge domination, ew, ez ∈S , a contradiction to the selection of S .
Lemma 3.9: If T ∈ ℑ with at least three edges, then T∈ℜ.
Proof: Let q= E(T) . Since T ∈ ℑ , we have γ t
'
(T ) = γ d (T ) . If diam (T ) ≤ 3 , then T is either a star or a
'
double star and γ t
'
(T ) = 2 = γ d (T ) . Therefore T ∈ ℜ . If diam (T ) ≥ 4 , assume that the result is true for all
'
T ' with E(T ) = q < q .
' '
trees
We prove the following Claim to prove above Lemma.
Claim 3.9.1: If there is an edge ea ∈E(T) such that T − ea contains at least two components P3 , then T ∈ ℜ .
= eb eb and ec ec are two components of T − ea . If T =T −{eb , eb}, then use D ' and S to
' ' ' '
Proof: Assume that P3
denote the γ t' -set of T ' containing ea and γ d - set of T , respectively. Since ea ∈ D ' , D ' U {eb } is a total edge
'
dominating set of T and hence γ t (T
' '
) ≥ γ t' (T) −1. Further since S is a γd
'
- set of T , S I {ea , eb , eb } = {ea , eb }
' '
by the definition of γd
'
- set. Clearly, S I E (T
'
) is a double edge dominating set of T ' and hence γ t' (T ' ) ≥
γ t' (T) −1=γ d (T) −1= S I E(T ') ≥ γ d (T ') . By using Theorem 1, we get γ t' (T ' ) = γ d' (T ')
' '
and so T '∈ ℑ .
By induction on T ' , T ' ∈ℜ . Now, since T is obtained from T ' by type - 2 operation, T ∈ ℜ .
By above claim, we only need to consider the case that, for the edge ea , T − ea has exactly one component P3 . Let
P = eu e ew ex ey ez ...er be a longest path in T having root at vertex r which is incident with er .
Clearly, C(ew) =C ' (ew) ≠ C1' (ew ) .By 3 of Lemma 6, C1' (ew ) = C3' (ew ) = φ . Hence P4 = eu e ew is a
component of T − ex . Let n be the number of components of P4 of S ( ex ) in T such that an end edge of every
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P4 is adjacent to ex . Suppose S ( ex ) in T has a component P4 with its support edge is adjacent to ex . Then it
consists of j number of P3 and k number of P2 components. By Lemma 6, m, j ∈ {0,1} and k ∈ {0,1, 2} .
Denoting the n components P4 of the sub graph S ( ex ) in T with one of its end edges is adjacent to an edge
ex in T by P4 = eui ei ewi , 1 ≤ i ≤ n . We prove that result according to the values of {m, j , k} .
Case 1: Suppose m = j = k =0. Then S ( ex ) = P4 ( n ) , n ≥ 1 in T. Further assume that
T ' = T − S [ ex ] , then 2 ≤ E (T '
) < q . Clearly, γ t
'
(T ' ) ≥ γ t' (T ) − 2n . Let S be a γ d - set of T such that S
'
contains as minimum number of edges of the sub graph S ( ex ) as possible. Then ex ∉ S and S I S[ex ] = 2n
by the definition of γd
'
- set. Clearly S I E (T ') is a double edge dominating set of T '
and
hence γ t (T ' ) ≥ γ t' (T ) − 2n = γ d (T ) −2n = S I E (T ' ) ≥ γ d (T ) . By Theorem 1, γ t' (T ' ) = γ d (T ' ) and
' ' ' '
S I E (T ' ) is a double edge dominating set of T ' . Hence T ' ∈ ℑ . By applying the inductive hypothesis to T ' ,
T ' ∈ℜ .
If n ≥ 2 , then it is obvious that T is obtained from T ' by type - 5 operation and hence T ∈ ℜ .
If n = 1 . Then S ( ex ) = P4 = eu e ew in T which is incident with x of an edge ex and
S I{eu , e, ew, ex} = {eu , ew} . To double edge dominate, ex , ey ∈ S and so ey ∈ S I E (T ' ) , which implies
that ey is in some γ d - set of T ' . Hence T is obtained from T ' by type-5 operation and T ∈ℜ .
'
Case 2: Suppose m ≠ 0 and by the proof of Lemma 6, m = 1 and j = k = 0 . Denote the component P4 of
S ( ex ) in T whose support edge is adjacent to ex in T by P4 = ea eb ec and if T ' = T − {ea , eb , ec } . Then,
clearly 3 ≤ E (T '
) ≤ q . Let S be a γ '
d - set of T which does not contain eb .
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Now we claim that ex is not in any γ t' - set of T ' . Suppose that T ' has a γ t' - set containing ex which is denoted
'
by D , then D ' U {eb } is a total edge dominating set of T . Clearly, γ t' (T ' ) ≥ γ t' (T ) −1 . Since eb ∉ S then
S I E (T ' ) is a double edge dominating set of T '. Hence
γ t' (T ' ) ≥ γ t' (T ) −1 = γ d (T ) −1 = S I E (T ' ) + 1≥ γ d (T ' ) + 1 ,
' '
which gives a contradiction to the fact that
γ t' (T ' ) ≤ γ d (T ' ) . This holds the claim and therefore T
'
can be obtained from T ' by type-4 operation.
Now we prove that T ' ∈ℜ . Let D ' be any γ t' - set of T ' . By above claim, ex ∉ D ' . Since D ' U {ex , eb } is a total
edge dominating set of T , γ t
'
(T ) ≥ γ (T ) − 2 . Further since e
'
t
'
b ∉ S , S I E (T ' ) is a double edge dominating
set of T ,
'
γ t' (T ' ) ≥ γ t' (T ) − 2 = γ d (T ) − 2 = S I E (T ' ) ≥ γ d (T ' ) .
' '
Therefore by Theorem 1, we get
γ t' (T ' ) = γ d (T ' ) ,
'
which implies that T ' ∈ ℑ . Applying the inductive hypothesis on T ' , T '∈ℜ and hence
T ∈ℜ .
Case 3: Suppose j ≠0 and by the proof of Lemma 6, m = k = 0 . Let i {
T ' =T − Un=1 eui ,ei ,ewi } . Clearly,
3 ≤ E (T ' ) < q and T is obtained from T ' by type - 3 operation.
We only need to prove that T '∈ℜ . Suppose D ' ⊂ E (T ' ) be a γ t' - set of T ' . Then
i ( { })
D' U Un=1 ei ,ewi is a total edge dominating set of T and hence. γ t' (T ' ) ≥ γ t' (T ) − 2n . Since T − ex has a
component P = ea eb , we can choose
3
S⊆ E(T) as a γd
'
- set of T containing ex . Then S I E(T ' ) is a γ d' -
set of T '
and hence γ d (T ) = S = 2n +
'
S I E (T ' ) ≥ 2n + γ d (T ' ) . Clearly, it follows that,
'
γ t' (T ' ) ≥ γ d (T ' ) . Therefore, by Theorem 1, we get, γ t' (T ' ) = γ d (T ' ) and hence T ' ∈ ℑ . Applying the inductive
' '
hypothesis on T ' , we get T '∈ℜ .
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Case 4: Suppose k ≠ 0 . Then by Lemma 6, k ∈{1,2} and so m = j = 0 . We claim k = 1 . If not, then k = 2 . We
denote the two components P2 of S ( ex ) by ex and ex in T . Let T ' = T − ex . Clearly, γ t' (T ' ) = γ t' (T ) . Let S
' '' ''
be a γd
'
- set of T containing {e w1 }
, ew2 ,..., ewn . By Observation 2, {ex , ex } ⊆ S . Since S I E(T ) is a double
' '' '
edge dominating set of T' with S I E(T ' ) = γ d (T) −1,
'
we have
γ t' (T ' ) = γ t' (T ) = γ d (T ) > γ d (T ) −1 ≥ γ d (T ' ) , which is a contradiction to the fact that γ t' (T ' ) ≤ γ d (T ' ) .
' ' ' '
Sub case 4.1: For n ≥ 2 . Suppose T
'
i {
=T − Un=1 eui ,ei ,ewi } . Then T is obtained from T '
by type - 3 operation.
Now by definition of γ t' - set and γd -
'
set, it is easy to observe that γ t (T
' '
) + 2 (n − 1) = γ t' (T ) and
γ d (T ' ) + 2(n − 1) = γ d (T )
' '
. Hence γ t' (T ' ) = γ d (T ' ) and T ' ∈ ℑ .
'
Applying the inductive hypothesis on T ',
T ' ∈ℜ and hence T ∈ ℜ .
Sub case 4.2: For n = 1 . Denote the component P2 of S ( ex ) by ex in T . Suppose S ( e y ) − S [ ex ] has a
'
component H∈{P , P , P } in T , then T ' = T − S [ ex ] . Therefore we can easily check that T
3 4 6 is obtained from
T ' by type-6 operation. Now by definition of γd -
'
set, γ d (T ' ) + 3 = γ d (T ) .
' '
For any γ t' '
- set D of T ',
D ' U {e , ew , ex } is a total edge dominating set of T . Clearly, γ t' (T ' ) ≥ γ t' (T ) − 3 = γ d (T ) − 3 = γ d (T ' ) . By
' '
Theorem 1, we get γ t' (T ' ) = γ d (T ' ) and T ' ∈ ℑ .
'
Applying inductive hypothesis on T , T ∈ ℜ and hence
' '
T ∈ℜ .
Now if the sub graph S ( ey ) − S [ ex ] has no components P3 , P4 or P6 . Then we consider the structure
S ( ey ) in T . By above discussion, S ( ey ) consists of a component P6 = eu e ew ex ex and g number of
'
of
components of P2 , denoted by {e , e ,..., e } . Assume l = 2 . Then, let T
1 2 g
'
= T − S ey . It can be easily checked
that γ t' (T ' ) + 4 ≥ γ d (T ) = γ d (T ' ) + 5 ,
' '
which is a contradiction to the fact that γ t' (T ' ) ≤ γ d (T ' ) .
'
Hence
g ≤ 1.
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Suppose T ' = T − {eu , ex } . Here we can easily check that γ t' (T ' ) + 1 = γ t' (T ) . Let S
'
be a γ d - set of
'
T such that S contains as minimum edges of S ey as possible and S I S [ex ] = {eu , ew ex } . Then
'
S ' = ( S − {eu , ew ex }) U {e, ex }
'
is a double edge dominating set of T '.
Therefore γ t
'
(T ' ) = γ t' (T ) −1 = γ d (T ) −1 = S ' ≥ γ d (T ' ) , which implies that γ t' (T ' ) = γ d (T ' ) where S ' is a
' ' '
double edge dominating set of T ' . Hence T ' ∈ ℑ . Applying inductive hypothesis to T ' , T ' ∈ ℜ .
If g = 0 , then degT ( ey ) = 2 . Since ex ∉S , to double edge dominate ey , ey ∈S . Therefore
ey is in the double edge dominating set D ' of T ' . Hence T is obtained from T ' by type-1 operation. Thus
T ∈ℜ .
If g = 1 , then degT ( ey ) = 3 . Since ex ∉S to double edge dominate ey , we have ey ∉S and
ez ∈S , by the selection of S . Therefore ez is in the double edge dominating set S ' of T ' . Hence T is obtained
from T by type-1operation. Thus T ∈ ℜ .
'
By above all the Lemmas, finally we are now in a position to give the following main characterization.
Theorem 3.10: ℑ = ℜ U { P3}
References
S. Arumugan and S. Velammal, (1997), “Total edge domination in graphs”, Ph.D Thesis, Manonmaniam Sundarnar
University, Tirunelveli, India.
F. Harary, (1969), “Graph theory”, Adison-wesley, Reading mass.
F. Harary and T. W. Haynes, (2000), “Double domination in graph”, Ars combinatorics, 55, pp. 201-213.
T.W. Haynes, S. T. Hedetniemi and P. J. Slater, (1998), “Fundamentals of domination in graph”, Marcel Dekker,
New York.
T. W. Haynes, S. T. Hedetniemi and P. J. Slater, (1998), “Domination in graph, Advanced topics”, Marcel Dekker,
New York.
Biography
A. Dr.M.H.Muddebihal born at: Babaleshwar taluk of Bijapur District in 19-02-1959.
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15. Mathematical Theory and Modeling www.iiste.org
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Vol.2, No.5, 2012
Completed Ph.D. education in the field of Graph Theory from Karnataka. Currently working as a Professor in
Gulbarga University, Gulbarga-585 106 bearing research experience of 25 years. I had published over 50 research
papers in national and international Journals / Conferences and I am authoring two text books of GRAPH THEORY.
B. A.R.Sedamkar born at: Gulbarga district of Karnataka state in 16-09-1984.
Completed PG education in Karnataka. Currently working as a Lecturer in Government Polytechnic Lingasugur,
Raichur Dist. from 04 years. I had published 10 papers in International Journals in the field of Graph Theory.
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