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.
International journal of applied sciences and innovation vol 2015 - no 2 - ...sophiabelthome
This document discusses domatic subdivision stable and just excellent graphs. It begins by introducing key concepts related to domination in graphs such as domatic number, dominating sets, and private neighborhood. It then defines domatic subdivision stable graphs as graphs where the domatic number is unchanged after any edge subdivision. Just excellent graphs are defined as graphs where every vertex is contained in a unique minimum dominating set. The document proves several results about the relationships between these graph properties, including that just excellent graphs are not domatic subdivision stable, and that the domatic number decreases to 2 after any edge addition between vertices in the same dominating set of a just excellent graph with domatic number 3.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
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
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.
Connected Total Dominating Sets and Connected Total Domination Polynomials of...iosrjce
Let G = (V, E) be a simple graph. A set S of vertices in a graph G is said to be a total dominating set
if every vertex v V is adjacent to an element of S. A total dominating set S of G is called a connected total
dominating set if the induced subgraph <s> is connected. In this paper, we study the concept of connected total
domination polynomials of the star graph Sn and wheel graph Wn. The connected total domination polynomial of
a graph G of order n is the polynomial Dct(G, x) =
ct
n
i=γ (G)
ct
i
d (G, i) x , where dct(G, i) is the number of
connected total dominating set of G of size i and ct(G) is the connected total domination number of G. We
obtain some properties of Dct(Sn, x) and Dct(Wn, x) and their coefficients. Also, we obtain the recursive formula
to derive the connected total dominating sets of the star graph Sn and the Wheel graph Wn
The document discusses studies on petal graphs. It begins with Angel Sophia submitting her dissertation to Manonmaniam Sundaranar University for her Master's degree in Mathematics. The dissertation is about studies on petal graphs under the guidance of her advisor Dr. G. Easwaraprasad. It includes an acknowledgement, contents, and introduction chapters to set up the topic of studying the properties and behaviors of petal graphs.
The document summarizes research on the energy of graphs. It defines the energy of a graph as the sum of the absolute values of its eigenvalues. It shows that for any positive epsilon, there exist infinitely many values of n for which a k-regular graph of order n exists whose energy is arbitrarily close to the known upper bound for k-regular graphs. It also establishes the existence of equienergetic graphs that are not cospectral. Equienergetic graphs have the same energy even if they do not have the same spectrum of eigenvalues.
A study on connectivity in graph theory june 18 pdfaswathymaths
This document provides an introduction and two chapters on connectivity in graphs. The introduction discusses the history and applications of graph theory. Chapter 1 defines key concepts related to connectivity such as bridges, cut vertices, and k-connectivity. It presents theorems characterizing when an edge is a bridge and when a graph is a tree. Chapter 2 discusses applications of connectivity in graphs.
International journal of applied sciences and innovation vol 2015 - no 2 - ...sophiabelthome
This document discusses domatic subdivision stable and just excellent graphs. It begins by introducing key concepts related to domination in graphs such as domatic number, dominating sets, and private neighborhood. It then defines domatic subdivision stable graphs as graphs where the domatic number is unchanged after any edge subdivision. Just excellent graphs are defined as graphs where every vertex is contained in a unique minimum dominating set. The document proves several results about the relationships between these graph properties, including that just excellent graphs are not domatic subdivision stable, and that the domatic number decreases to 2 after any edge addition between vertices in the same dominating set of a just excellent graph with domatic number 3.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
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
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.
Connected Total Dominating Sets and Connected Total Domination Polynomials of...iosrjce
Let G = (V, E) be a simple graph. A set S of vertices in a graph G is said to be a total dominating set
if every vertex v V is adjacent to an element of S. A total dominating set S of G is called a connected total
dominating set if the induced subgraph <s> is connected. In this paper, we study the concept of connected total
domination polynomials of the star graph Sn and wheel graph Wn. The connected total domination polynomial of
a graph G of order n is the polynomial Dct(G, x) =
ct
n
i=γ (G)
ct
i
d (G, i) x , where dct(G, i) is the number of
connected total dominating set of G of size i and ct(G) is the connected total domination number of G. We
obtain some properties of Dct(Sn, x) and Dct(Wn, x) and their coefficients. Also, we obtain the recursive formula
to derive the connected total dominating sets of the star graph Sn and the Wheel graph Wn
The document discusses studies on petal graphs. It begins with Angel Sophia submitting her dissertation to Manonmaniam Sundaranar University for her Master's degree in Mathematics. The dissertation is about studies on petal graphs under the guidance of her advisor Dr. G. Easwaraprasad. It includes an acknowledgement, contents, and introduction chapters to set up the topic of studying the properties and behaviors of petal graphs.
The document summarizes research on the energy of graphs. It defines the energy of a graph as the sum of the absolute values of its eigenvalues. It shows that for any positive epsilon, there exist infinitely many values of n for which a k-regular graph of order n exists whose energy is arbitrarily close to the known upper bound for k-regular graphs. It also establishes the existence of equienergetic graphs that are not cospectral. Equienergetic graphs have the same energy even if they do not have the same spectrum of eigenvalues.
A study on connectivity in graph theory june 18 pdfaswathymaths
This document provides an introduction and two chapters on connectivity in graphs. The introduction discusses the history and applications of graph theory. Chapter 1 defines key concepts related to connectivity such as bridges, cut vertices, and k-connectivity. It presents theorems characterizing when an edge is a bridge and when a graph is a tree. Chapter 2 discusses applications of connectivity in graphs.
This document discusses orthogonal subspaces and inner products in advanced engineering mathematics. It defines the inner product of two vectors u and v in Rn as the transpose of u dotted with v, which results in a scalar. Two vectors are orthogonal if their inner product is 0. An orthogonal basis for a subspace W is a basis for W that is also an orthogonal set. The document also discusses orthogonal complements, projections, and inner products on function spaces.
A study on connectivity in graph theory june 18 123easwathymaths
This document provides an introduction to connectivity of graphs. It begins with definitions of terms like bridges, cut vertices, connectivity, and edge connectivity. It then presents several theorems about when edges are bridges and vertices are cut vertices. It proves properties of trees related to cut vertices. The document establishes relationships between vertex and edge connectivity. It introduces the concepts of k-connectivity and discusses properties of complete graphs and trees in relation to connectivity.
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
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.
IOSR Journal of Mathematics(IOSR-JM) is an open access 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.
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.
IRJET- On Strong Domination Number of JumpgraphsIRJET Journal
This document discusses strong domination in jump graphs. It begins with definitions of domination, strong domination, and related graph parameters. It then presents several theorems about strong domination numbers and sets in graphs. Specifically, it proves that vertices of maximum degree must be in a strong dominating set, and provides bounds on the strong domination number under certain conditions. It concludes by discussing applications of strong domination to network security.
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.
Nelly Litvak presents a document on degree-degree dependencies in random graphs with heavy-tailed degrees. She discusses Newman's assortativity coefficient ρ(G) which is a measure of correlations between the degrees of connected nodes. Positive values indicate assortative mixing where high degree nodes connect to other high degree nodes, while negative values indicate disassortative mixing. She reviews that technological and biological networks are typically disassortative while social networks are assortative. Litvak then presents theorems showing that in power law graphs with γ ∈ (1,3), the assortativity coefficient converges to a non-negative value, so these graphs are never strongly disassortative. She also discusses
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document defines and discusses the concept of paired triple connected domination number of a graph. It begins by reviewing existing concepts like domination number, connected domination number, and triple connected domination number. It then introduces the new concept of a paired triple connected dominating set as a triple connected dominating set where the induced subgraph also has a perfect matching. The paired triple connected domination number is defined as the minimum cardinality of such a set. The document explores properties of this number and its relationship to other graph parameters. Examples are provided to illustrate the definitions.
The document presents results on determining the metric dimension of t-fold wheel graphs. It defines t-fold wheel graphs and provides previous results on the metric dimension of other graphs. The main result is a theorem stating that for t-fold wheel graphs, the metric dimension is t+1 if the number of vertices n is 3, 4, or 5, and (n+t-2)/2 if n is greater than or equal to 6. The proof considers two cases based on the value of n and shows that sets of certain vertices are both resolving sets and cannot be made smaller.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
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.
Multilayerity within multilayerity? On multilayer assortativity in social net...Moses Boudourides
This document discusses assortative mixing and partitions in multilayer networks. It defines graph partitions, ordering of partitions, self-similarity of partitions, and how partitions can be viewed as enumerative attribute assignments. It then defines the assortativity coefficient of a partition to measure how strongly vertices in the same group of a partition are connected. Finally, it discusses how to measure assortativity between two different partitions of a graph by focusing on their intersection partition.
The document defines symmetric groups and discusses their properties. Some key points:
- A symmetric group is the group of all permutations of a finite set under function composition.
- Symmetric groups of finite sets behave differently than those of infinite sets.
- The symmetric group Sn of degree n is the set of all permutations of the set {1,2,...,n}.
- Sn is a finite group under permutation composition. Subgroups include the alternating group An of even permutations.
- Examples discussed include S2, the Klein four-group, and S3, which is non-abelian with cyclic subgroups.
A NEW APPROACH TO M(G)-GROUP SOFT UNION ACTION AND ITS APPLICATIONS TO M(G)-G...ijscmcj
In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group theory.
Topics of Complex Social Networks: Domination, Influence and AssortativityMoses Boudourides
The document summarizes topics related to complex social networks, including domination, influence, and assortativity. It begins by defining dominating sets in graphs and their properties such as minimal dominating sets. It describes the complexity of computing dominating sets and provides algorithms. It then discusses egocentric subgraphs induced by dominating sets and the classification of vertices as private or public alters. Finally, it introduces notation used to describe edges between dominating sets, private alters, and public alters.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
This document summarizes a research paper on improving traffic detection algorithms using an extended floating car data (xFCD) system. The xFCD system collects data from vehicles including location, speed, direction and visual data from a forward-facing camera. It is tested under different lighting and traffic conditions. The paper investigates using xFCD data and information from road sensors to construct a hybrid model characterizing traffic states. A traffic detection algorithm is proposed to improve network performance metrics like throughput, delivery ratio and packet delay. Simulation results show the proposed approach improves these metrics compared to existing methods.
The document discusses a lunch meeting between two security industry recruitment consultants, Mike Hurst and Graham Bassett, about job interviews and changing jobs. Some key points:
- January is a popular time for people to look for or change jobs as they have had time to reflect over the holidays.
- If unhappy in a current role, the grass may seem greener elsewhere but it's important to do research to understand a new role and company culture fully before making a change.
- CVs and online profiles like LinkedIn are still important in the hiring process but interviews provide valuable additional insights beyond what is written.
- The hiring process may involve multiple stages like phone screens, interviews and presentations for senior
International Journal of Engineering and Science Invention (IJESI)inventionjournals
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
International Journal of Engineering and Science Invention (IJESI)inventionjournals
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
This document discusses orthogonal subspaces and inner products in advanced engineering mathematics. It defines the inner product of two vectors u and v in Rn as the transpose of u dotted with v, which results in a scalar. Two vectors are orthogonal if their inner product is 0. An orthogonal basis for a subspace W is a basis for W that is also an orthogonal set. The document also discusses orthogonal complements, projections, and inner products on function spaces.
A study on connectivity in graph theory june 18 123easwathymaths
This document provides an introduction to connectivity of graphs. It begins with definitions of terms like bridges, cut vertices, connectivity, and edge connectivity. It then presents several theorems about when edges are bridges and vertices are cut vertices. It proves properties of trees related to cut vertices. The document establishes relationships between vertex and edge connectivity. It introduces the concepts of k-connectivity and discusses properties of complete graphs and trees in relation to connectivity.
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
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.
IOSR Journal of Mathematics(IOSR-JM) is an open access 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.
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.
IRJET- On Strong Domination Number of JumpgraphsIRJET Journal
This document discusses strong domination in jump graphs. It begins with definitions of domination, strong domination, and related graph parameters. It then presents several theorems about strong domination numbers and sets in graphs. Specifically, it proves that vertices of maximum degree must be in a strong dominating set, and provides bounds on the strong domination number under certain conditions. It concludes by discussing applications of strong domination to network security.
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.
Nelly Litvak presents a document on degree-degree dependencies in random graphs with heavy-tailed degrees. She discusses Newman's assortativity coefficient ρ(G) which is a measure of correlations between the degrees of connected nodes. Positive values indicate assortative mixing where high degree nodes connect to other high degree nodes, while negative values indicate disassortative mixing. She reviews that technological and biological networks are typically disassortative while social networks are assortative. Litvak then presents theorems showing that in power law graphs with γ ∈ (1,3), the assortativity coefficient converges to a non-negative value, so these graphs are never strongly disassortative. She also discusses
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document defines and discusses the concept of paired triple connected domination number of a graph. It begins by reviewing existing concepts like domination number, connected domination number, and triple connected domination number. It then introduces the new concept of a paired triple connected dominating set as a triple connected dominating set where the induced subgraph also has a perfect matching. The paired triple connected domination number is defined as the minimum cardinality of such a set. The document explores properties of this number and its relationship to other graph parameters. Examples are provided to illustrate the definitions.
The document presents results on determining the metric dimension of t-fold wheel graphs. It defines t-fold wheel graphs and provides previous results on the metric dimension of other graphs. The main result is a theorem stating that for t-fold wheel graphs, the metric dimension is t+1 if the number of vertices n is 3, 4, or 5, and (n+t-2)/2 if n is greater than or equal to 6. The proof considers two cases based on the value of n and shows that sets of certain vertices are both resolving sets and cannot be made smaller.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
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.
Multilayerity within multilayerity? On multilayer assortativity in social net...Moses Boudourides
This document discusses assortative mixing and partitions in multilayer networks. It defines graph partitions, ordering of partitions, self-similarity of partitions, and how partitions can be viewed as enumerative attribute assignments. It then defines the assortativity coefficient of a partition to measure how strongly vertices in the same group of a partition are connected. Finally, it discusses how to measure assortativity between two different partitions of a graph by focusing on their intersection partition.
The document defines symmetric groups and discusses their properties. Some key points:
- A symmetric group is the group of all permutations of a finite set under function composition.
- Symmetric groups of finite sets behave differently than those of infinite sets.
- The symmetric group Sn of degree n is the set of all permutations of the set {1,2,...,n}.
- Sn is a finite group under permutation composition. Subgroups include the alternating group An of even permutations.
- Examples discussed include S2, the Klein four-group, and S3, which is non-abelian with cyclic subgroups.
A NEW APPROACH TO M(G)-GROUP SOFT UNION ACTION AND ITS APPLICATIONS TO M(G)-G...ijscmcj
In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group theory.
Topics of Complex Social Networks: Domination, Influence and AssortativityMoses Boudourides
The document summarizes topics related to complex social networks, including domination, influence, and assortativity. It begins by defining dominating sets in graphs and their properties such as minimal dominating sets. It describes the complexity of computing dominating sets and provides algorithms. It then discusses egocentric subgraphs induced by dominating sets and the classification of vertices as private or public alters. Finally, it introduces notation used to describe edges between dominating sets, private alters, and public alters.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
This document summarizes a research paper on improving traffic detection algorithms using an extended floating car data (xFCD) system. The xFCD system collects data from vehicles including location, speed, direction and visual data from a forward-facing camera. It is tested under different lighting and traffic conditions. The paper investigates using xFCD data and information from road sensors to construct a hybrid model characterizing traffic states. A traffic detection algorithm is proposed to improve network performance metrics like throughput, delivery ratio and packet delay. Simulation results show the proposed approach improves these metrics compared to existing methods.
The document discusses a lunch meeting between two security industry recruitment consultants, Mike Hurst and Graham Bassett, about job interviews and changing jobs. Some key points:
- January is a popular time for people to look for or change jobs as they have had time to reflect over the holidays.
- If unhappy in a current role, the grass may seem greener elsewhere but it's important to do research to understand a new role and company culture fully before making a change.
- CVs and online profiles like LinkedIn are still important in the hiring process but interviews provide valuable additional insights beyond what is written.
- The hiring process may involve multiple stages like phone screens, interviews and presentations for senior
International Journal of Engineering and Science Invention (IJESI)inventionjournals
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
International Journal of Engineering and Science Invention (IJESI)inventionjournals
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
Este documento contiene un registro de las observaciones del huerto escolar de las clases 2A y 2B entre el 30 de septiembre y el 25 de octubre de 2013. Cada entrada incluye la fecha, la temperatura, la cantidad de precipitación y notas sobre el estado de las plantas, como el crecimiento de los ajos y las flores en las berenjenas.
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.
This candidate has over 20 years of experience in human resources, benefits administration, and legal work. They are bilingual in English and Spanish with experience in various HR systems and office software. Currently they work as an Enrollment Implementation Analyst at Aflac where they help implement benefit enrollment systems for large company accounts.
SMS Systems Maintenance Services is a global IT services company with over 2000 engineers and 1000 service affiliates across locations in India, France, and the US. They provide 24/7 infrastructure support and professional services to multinational companies, including cloud providers, through their multi-vendor support, data center services, and project management.
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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
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1. International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org Volume 2Issue 12ǁ December 2013 ǁ PP.32-37
Graphs In Which Upper Strong Efficient Domination Number
Equals The Independence Number
1,
N.Meena , 2, A.Subramanian , 3, V.Swaminathan
1,
Department of Mathematics, The M.D.T Hindu College, Tirunelveli 627 010, India.
Dean, College Development Council, Manonmaniam Sundaranar University, Tirunelveli 627 012, India.
3,
Ramanujan Research Center, Department of Mathematics, Saraswathi Narayanan College, Madurai 625 022.
2,
ABSTRACT: Let G = (V, E) be a simple graph. A subset S of V(G) is called a strong (weak) efficient
dominating set of G if for every v V(G), │Ns[v]∩S│ = 1 (│Nw[v]∩S│ = 1), where Ns(v) = { u V(G): uv
E(G), deg u ≥ deg v }and Nw(v) = { u V(G) : uv E(G), deg v ≥ deg u}, Ns[v] = Ns(v) {v}, (Nw[v] = Nw(v)
{v}). The minimum cardinality of a strong (weak) efficient dominating set is called the strong (weak) efficient
domination number of G and is denoted by γse(G) (γwe(G)). A graph is strong efficient if there exists a strong
efficient dominating set of G. The graphs which have full degree vertex admit strong efficient dominating set.
Not all graphs admit strong efficient dominating sets. The complete bipartite graph Km,n, m,n 2, doesn’t admit
strong efficient dominating set. A subset S of V(G) is called an efficient dominating set of G if N[v] S = 1
for all vertices vV(G). If G has an efficient dominating set, then the cardinality of any efficient dominating set
equals the domination number (G) [1]. In particular all efficient dominating sets of G have the same
cardinality. But this is not true in the case of strong efficient domination. The maximum cardinality of any
strong efficient dominating set of G is called the upper strong efficient domination number of G and is denoted
by se (G). A set of vertices of G is said to be independent if no two of them are adjacent. The maximum number
of vertices in any independent set of G is called the independence number of G and is denoted by
. Let
and
be two graphs with vertex sets and
and edge sets
and
respectively. The Cartesian product
is defined to be the graph whose vertex set is
and edge set is
. In this paper, we introduce the
concept of perfect graphs of upper strong efficient domination and maximum independent sets. That is, graphs
G in which se (G) =
. A study of such graphs is made. Also strong efficient domination in product graphs
is discussed.
KEY WORDS: Strong efficient domination number, strong dominating set, independent strong dominating set.
AMS Subject Classification (2010): 05C69.
I.
INTRODUCTION
Throughout this paper, we consider finite, undirected, simple graphs. Let G = (V,E) be a simple graph.
The degree of any vertex u in G is the number of edges incident with u and is denoted by deg u. The minimum
and maximum degree of a vertex is denoted by (G) and (G) respectively. A vertex of degree 0 in G is called
an isolated vertex and a vertex of degree 1 is called a pendant vertex. A subset S of V(G) of a graph G is called a
dominating set if every vertex in V(G) S is adjacent to a vertex in S. The domination number (G) is the
minimum cardinality of a dominating set of G. E. Sampathkumar and L.Pushpalatha introduced the concepts of
strong and weak domination in graphs [8]. A subset S of V(G) is called a strong dominating set of G if for
every v V – S there exists uS such that u and v are adjacent and deg u ≥ deg v. The strong domination
number s(G) is the minimum cardinality of a strong dominating set of G and the strong domination number
s(G) is the maximum cardinality of a strong dominating set of G. The independent strong domination number
is(G) is the minimum cardinality of an independent strong dominating set of G. The strong domination and
independent numbers were studied in [3, 4, 6, 7, 8]. The upper independent strong domination number s(G) is
the maximum cardinality of an independent strong dominating set of G. The maximum number of vertices in
any independent set of G is called the independence number of G and it is denoted by
The inequality
chain s (G)
is (G)
(G)
(G)
(G)
holds in any graph admitting strong efficient
se
s
s
dominating set. It has been proved in [5], that there are graphs in which strict inequality holds in the chain. In
this paper, we introduce the concept of perfect graphs of upper strong efficient domination and maximum
independent sets. That is, graphs G in which se (G) =
. A study of such graphs is made. Strong efficient
www.ijesi.org
32 | Page
2. Graphs In Which Upper Strong Efficient Domination...
domination in products of graphs is also discussed. For all graph theoretic terminologies and notations, we
follow Harary [2].
II.
MAIN RESULTS
Definition 2.1: Let G = (V, E) be a simple graph. A subset S of V(G) is called a strong (weak) efficient
dominating set of G if for every v V(G),│Ns[v]∩S│= 1 (│Nw[v]∩S│= 1), where Ns(v) = { u V(G) : uv
E(G), deg u ≥ deg v } and Nw(v) = { u V(G) : uv E(G), deg v ≥ deg u}, Ns[v] = Ns(v) {v} (Nw[v] = Nw(v)
{v}). The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak)
efficient domination number of G and is denoted by se(G) ( we(G) ). A graph G is strong efficient if there exists
a strong efficient dominating set of G.
Definition 2.2: A subset S of V(G) which is strong efficient and whose cardinality is β0(G) is called
independence number preserving strong efficient dominating set of G.
Remark 2.3: Since any strong efficient dominating set is independent, Γ se(G) ≤ β0(G). Therefore if there exists
an independence number preserving strong efficient dominating set of G, then such a set has cardinality Γ se(G)
and Γse(G) = β0(G).
Remark 2.4: Suppose there exists a graph G such that se(G) = β0(G). Then se(G) = Γse(G) = β0(G).
Example 2.5: Let G = P5. {v1, v3, v5} is the unique strong efficient dominating set which is an independent set
of maximum cardinality also. Therefore se(G) = β0(G).
Theorem 2.6: Let Gn (n ≥ 3) be a graph obtained by attaching a vertex of K n with the central vertex of K1, n-1
by an edge. Then Γse(Gn) = β0(Gn).
Proof: Let V(Kn) = {v1, v2, … , vn} and V(K1, n-1) = {u, u1, u2, … , un-1}. In Gn, u is attached with any vertex,
say vi, of Kn. Therefore deg u = deg vi = n = (Gn). For j = 1 to n consider the sets Sj = {u, vj}, j i. Then u
strongly dominates u1, u2, … , un-1 and vi. The vertex vj strongly dominates all the vertices of Kn other than vi,
since deg vj < deg vi, j i. Hence Sj, j = 1 to n are strong efficient dominating sets of Gn and │Sj│ = 2. Consider
the set S = {vi, u1, u2, … , un-1}. v1 strongly dominates the vertices of Kn and u. The vertices u1, u2, … , un-1
dominate themselves. Clearly S is also a strong efficient dominating set of Gn and │S│ = n. The sets Sj s and S
are the independent and S is of maximum cardinality. Since any strong efficient dominating set of Gn must
contain either u or v1, no other strong efficient dominating sets exist. Therefore se(Gn) = 2 and Γse(Gn) = β0(Gn)
= n where (n ≥ 3). That is S is the independence number preserving strong efficient dominating set of G n.
Remark 2.7: In the above theorem, when n = 1, Gn is P2, se(Gn) = β0(Gn) =1.
When n = 2, Gn is P4, se(Gn) = β0(Gn) = 2.
Theorem 2.8: A graph G doesnt admit a strong efficient dominating set if G has exactly two vertices of degree
(G) such that d (u,v) = 2.
Proof: Let G = ( V, E ) be a simple graph. Suppose G admits a strong efficient dominating set S. Let u, v be the
vertices of degree (G) such that d (u, v) = 2. Therefore there exists a vertex w V (G) adjacent with u and v.
Since u and v are not adjacent, u, v S. Therefore │Ns[w] S│ 2, a contradiction. Therefore G doesn’t admit
a strong efficient dominating set if G has exactly two vertices of degree (G) such that d (u, v) = 2.
Theorem 2.9: Let G = Pk where k = 2n + 1, n > 2. Attach a vertex u with G and make u adjacent with every
vertex of an independent set {v2, v4, ...v2n} of G. The resulting graph H is strong efficient and se(H) = β0(H).
Proof: Let V(G) = {v1, v2, v3, … , v2n+1}. In H, u is made adjacent with v2, v4, … , v2n. Since deg u = n = (G),
u is the only maximum degree vertex of H. Since deg vi > deg vj for all i (2, 4, … , 2n) and j (1, 3, 5, … ,
2n+1). vj cannot strongly dominate vi. Therefore T = {u, v1, v3, v5, … v2n-1, v2n+1} is the unique strong efficient
dominating set of H and T is also the maximum independent set of H. Therefore H is strong efficient and se(H)
= β0(H).
Remark 2.10: (i) When n = 2, G is P5. If u is made adjacent with an independent set {v2, v4} of G, then v2
and v4 are the only maximum degree vertices at a distance 2 in the resulting graph is H. By theorem 2.8, H is
non strong efficient.
(ii) When n = 1, G is P3. u is made adjacent with an independent set {v2}. In the resulting graph H, v2 is the full
degree vertex of H and hence se(H) = 1. But β0(H) = 3. Therefore se(H) β0(H).
www.ijesi.org
33 | Page
3. Graphs In Which Upper Strong Efficient Domination...
Theorem 2.11: Let G = C2n, n ≥ 3 . Attach a vertex u with G and make u adjacent with every vertex of a
maximum independent set of G. Then the resulting graph H is strong efficient and se(H) = β0(H).
Proof: Let V(G) = {v1, v2, … , v2n}. Let the new vertex u be adjacent with v2, v4,.. v2n. Therefore u is the only
maximum degree vertex of the resulting graph H. The set S = {v1, v3, v5,...v2n-1} of non neighbours of u is
independent and for any vi belongs to S, deg vi is less than that of any vertex of N[u]. Therefore T = {u, v1, v3,
… , v2n-1} is the unique strong efficient dominating set of H and T is an independent set of maximum
cardinality. Hence se(H) = β0(H) = n + 1.
Remark 2. 12: When n = 2, G = C4. If u is made adjacent either with v2, v4 or with v1, v3, then the resulting
graph H is non strong efficient, since distance between the maximum degree vertices is 2, by theorem 2.8.
Definition 2.13 [2]: A rooted tree has one point, its root v distinguished from the others. A rooted tree with root
of degree n can be regarded as a configuration whose figures are the n rooted trees obtained on removing the
root. G – v are the constituent rooted trees.
Theorem 2.14: Let u be the root of a rooted tree G of degree n, n ≥ 3. Let one of the constituent rooted tree be
K1, n-1 and remaining be K1,r where r < n – 1. Then G is strong efficient and Γse(G) = β0(G).
Proof: Let v1, v2, … , vn be the vertices adjacent with u. Without loss of generality, let deg v1 = n – 1 and
deg vi = n – 2, 2 ≤ i ≤ n. Therefore u and v1 are the only maximum degree vertices. Let v11, v12, … , v1n-1 be the
vertices adjacent with v1, and vi1, vi2, … , vin-2 be the vertices adjacent with vi, 2 ≤ i ≤ n. u strongly dominates v1,
v2, … , vn, and v11, v12, … , v1n-1, vi1, vi2, … , vin-2, 2 ≤ i ≤ n are independent of u. Let S1 = {v1, v2, … , vn} and
S2 = {u, v11, v12, … , v1n-1, v21, v22, … , v2(n-2), … , vn1, vn2, … , vn(n-2)}. Clearly S1 and S2 are strong efficient
dominating sets. No other strong efficient dominating set without u or v1 exist. │S1│ = n and │S2│= 1 + (n - 1)
+ (n - 1)(n - 2) = n2 – 2n + 2, where n 3. se(G) = n and Γse(G) = n2 – 2n + 2, and S2 is the maximum
independent set of G. Thus Γse(G) = β0(G). Therefore S2 is the independence number preserving strong efficient
dominating set of G.
III.
STRONG EFFICIENT DOMINATION IN PRODUCT GRAPHS.
Definition 3.1: Let G1 and G2 be two graphs with vertex sets V1 and V2 and edge sets E1 and E2 respectively.
The Cartesian product G1 □ G2 is defined to be the graph whose vertex set is V1 V2 and edge set is {(u1, v1),
(u2, v2) / either u1 = u2 and v1v2 E2 or v1 = v2 and u1u2 E1}.
Theorem 3.2: P3 □ P3 is strong efficient.
Proof: Let V(P3) = {u1, u2, u3}. V(P3) = {v1, v2, v3}. (u2, v2) is the only maximum degree vertex of P 3 □ P3.
(u2, v2) strongly uniquely dominates (u1, v2), (u2, v1), (u2, v3), (u3, v2). The vertices (u1, v1), (u1, v3), (u3, v1),
(u3, v3) are independent of (u2, v2). Hence {(u2, v2), (u1, v1), (u1, v3), (u3, v1), (u3, v3)} is the unique se- set of G.
Therefore se(P3 □ P3) = 5.
Theorem 3.3: P2 □ P3 is not strong efficient.
Proof: Let V(P2) = {u1, u2}. V(P3) = {v1, v2, v3}. Suppose S is a strong efficient dominating set of G. Either
(u1, v2) belongs to S or (u2, v2) belongs to S, since they are the only maximum degree vertices. Suppose (u1, v2)
belongs to S. (Proof is similar if (u2, v2) belongs to S). In this case (u1, v1), (u1, v3), (u2, v2) are strongly
dominated by (u1, v2) in S. Therefore (u2, v1) belongs to S. Then (u1, v1) is strongly dominated by (u1, v2) and
(u2, v1) in S, a contradiction. Therefore P2 □ P3 is not strong efficient.
Theorem 3.4:
P2 □ Pn is not strong efficient for any n ≥ 2.
Proof: For n = 3, the proof is given in the previous theorem. Suppose n ≥ 4. Let V(P2) = {u1, u2}. V(Pn) = {v1,
v2, ..., vn}. (u1, v1), (u2, v1), (u1, vn) and (u2, vn) are of degree two and all the other vertices of degree three.
Suppose S is a strong efficient dominating set of G. Then (u1, v2) belongs to S or (u1, v3) belongs to S or (u2, v2)
belongs to S. If (u1, v2) belongs to S, then (u2, v1) belongs to S leading to a contradiction. If (u 1, v3) belongs to
S, then (u2, v2) belongs to S leading to a contradiction. If (u2, v2) belongs to S, then (u1, v1) belongs to S leading
to a contradiction. For n = 2, P2 □ P2 = C4, which is not strong efficient. Therefore P2 □ Pn is not strong efficient
for any n ≥ 2.
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34 | Page
4. Graphs In Which Upper Strong Efficient Domination...
Theorem 3.5: K1,n □ P3 is strong efficient and se(K1,n □P3) = 2n + 1, n ≥ 1 .
Proof: Let V(K1,n) = {u1, u2, ..., un+1}. V(P3) = {v1, v2, v3}. V(K1,n □P3) = {(ui, vj)/1≤ i ≤ n+1, 1≤ j ≤ 3}. {(u1, v2),
(u2, v1), (u3, v1), ….. , (un+1, v1), (u2, v3), (u3, v3), …. , (un+1, v3)} is the unique se – set of K1,n □ P3. Therefore
se (K1,n □ P3) = 2n + 1. (Any se – set of a graph G must contain the vertex with maximum degree (G) if it is
unique). If S is a se – set of K1,n □ P3, then (u1, v2) which is of degree n+2 belongs to S. No neighbour of (u1, v2)
belongs to S, since S is independent. Hence (u1, v1), (u1, v3), (ui, v2), 2 ≤ i ≤ n+1, do not belong to S and they are
strongly dominated by (u1, v2). The remaining 2n vertices in K1,n □ P3 are independent and hence any se – set
must contain all these 2n vertices. Therefore se – set of K1,n □ P3 is unique.
Theorem 3.6: K1,n □ Pm is not strong efficient if m 3.
Proof Case (i): Let m ≥ 4. Let V(K1,n) = {u1, u2, … , un+1} and V(Pm) = {v1, v2, … , vm}. (u1, v1) and (u1, vm)
are of degree n+1. (u1, v2), … , (u1, vm-1) are of degree n+2. Let S be a strong efficient dominating set of G =
K1,n □ Pm. If (u1, v2) belongs to S then (u1, v4) does not belong to S. (Since otherwise (u 1, v3) will be strongly
dominated by 2 vertices of S) (u1, v3) also does not belong to S, since it is adjacent with (u 1, v2). Therefore (u2,
v3) belongs to S. In this case, (u2, v2) is strongly dominated by (u1, v2) and (u2, v3), a contradiction. Therefore
(u1, v2) S. Therefore (u1, v3) S. (Since (u1, v2) is strongly dominated only by (u1, v3)). Therefore (ui, v3) S,
2 ≤ i ≤ n+1. (u2, v2) belongs to S, since (u2,v2) is strongly dominated by (u1, v2) and (u2, v3) which do not belong
to S. In this case (u2, v3) is strongly dominated by two vertices of S namely (u2, v2) and (u1, v3), a contradiction.
Therefore K1,n □ Pm is not strong efficient.
Case (ii): Let m = 2. V(P2) = {v1, v2}. (u1, v1) and (u1, v2) are of degree n+1 and the remaining vertices are of
degree two. Suppose K1,n □ P2 is strong efficient. Let S be a strong efficient dominating set of K1,n □ P2. Then S
must contain either (u1, v1) or (u1, v2). If (u1, v1) S then (u1, v1) strongly dominates (ui, v1), i = 2, 3, … , n+1
and (u1, v2). Then (ui, v2), i = 2, 3, … , n+1 belong to S, since they are independent. Hence │Ns[(ui, v1) ] S│ =
│{(u1, v1), (ui, v2)}│ = 2 > 1 for all i = 2, 3, … , n+1, a contradiction. Proof is similar if (u1, v2) S. Therefore
K1,n □ P2 is not strong efficient.
Theorem 3.7:
C4n □ P2, n N is strong efficient.
Proof: Let V(C4n) = { u1, u2, … , un } and V(P2) = {v1, v2}.
deg (ui, v1) = deg (ui, v2) = 3 for all i = 1, 2, … , 4n.
S1 = {(u1, v1), (u3, v2), (u5, v1), (u7, v2), (u9, v1), (u11, v2), …. , (u4n-3, v1), (u4n-1, v2)}
S2 = {(u2, v1), (u4, v2), (u6, v1), (u8, v2), (u10, v1), (u12, v2), …. , (u4n-2, v1), (u4n, v2)}
S3 = {(u3, v1), (u5, v2), (u7, v1), (u9, v2), (u11, v1), (u13, v2), …. , (u4n-1, v1), (u4n-3, v2), (u1, v2)} and S4 = {(u4, v1),
(u6, v2), (u8, v1), (u10, v2), (u12, v1), …. , (u4n, v1), (u4n-2, v2), (u2, v2)}. S1, S2, S3 and S4 are the se – sets of
C4n □ P2, n N. Therefore C4n □ P2, n N is strong efficient.
Theorem 3.8: Cm □ P2, m 4n, n N is not strong efficient.
Proof: Case (i): m = 4n+1, n N. V(Cm) = {u1, u2, … , u4n+1} and V(P2) = {v1, v2}.
Suppose S is a strong efficient dominating set of C4n+1 □ P2. C4n+1 □ P2 is a 3 – regular graph. If (u1, v1) belongs
to S then (u1, v2), (u2, v1) and (u4n+1, v1) do not belong to S. (u2, v2) does not belong to S, since otherwise (u1, v2)
is strongly dominated by two elements (u1, v1) and (u2, v2) of S. Similarly (u3, v1) does not belong to S.
Therefore (u3, v2) belongs to S. Proceeding like this we get
(u5, v1), (u7, v2), …., (u4n-1, v2) belong to S.
(u4n-1, v2) strongly dominates (u4n-1, v1), (u4n-2, v2) and (u4n, v2). The vertices (u4n, v1), (u4n+1, v2) are not strongly
dominated by any vertex of S and also they are non adjacent vertices. Therefore (u4n, v1) and (u4n+1, v2) belong to
S. But (u4n, v2) is strongly dominated by three elements of S namely (u4n, v1), (u4n+1, v2) and (u4n-1, v2), a
contradiction. The case is similar if any other (ui, v1) or (ui, v2), 1 ≤ i ≤ 4n+1, belongs to S. Hence C4n+1 □ P2 is
not strong efficient.
Case (ii): m = 4n+2, n N. V(Cm) = {u1, u2, … , u4n+1, u4n+2}.
Suppose S is a strong efficient dominating set of C4n+2 □ P2. If (u1, v1) belongs to S, then as in case (i), (u3, v2),
(u5, v1), (u7, v2), …., (u4n-1, v2), (u4n+1, v1) belong to S. But (u4n+2, v1) is strongly dominated by two elements
(u4n+1, v1) and (u1, v1) of S, a contradiction and (u4n+2, v2) is not dominated by any element of S. Also (u4n+2, v2)
does not belong to S, since otherwise (u4n+2, v1) is strongly dominated by three elements namely (u4n+1, v1),
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5. Graphs In Which Upper Strong Efficient Domination...
(u4n+2, v2) and (u1, v1) of S, a contradiction. The case is similar if any other (ui, v1) or (ui, v2), 1 ≤ i ≤ 4n+2,
belongs to S. Therefore C4n+2 □ P2 is not strong efficient.
Case (iii): m = 4n+3, n N. V(Cm) = {u1, u2, … , u4n+1, u4n+2, u4n+3}.
Suppose S is a strong efficient dominating set of C4n+3 □ P2. If (u1, v1) belongs to S, then as in case (i), (u3, v2),
(u5, v1), (u7, v2), …., (u4n+1, v1), (u4n+3, v2) belong to S. (u4n+3, v1) is strongly dominated by two elements (u1, v1)
and (u4n+3, v2) of S, a contradiction. The case is similar if any other (u i, v1) or (ui, v2), 1 ≤ i ≤ 4n+3, belong to S.
Hence C4n+3 □ P2 is not strong efficient.
Remark 3.9: C3 □ P2 is not strong efficient.
Proof: Let V(C3) = {u1, u2, u3}. V(P2) = {v1, v2}. Suppose S is a strong efficient dominating set of C3 □ P2. deg
(ui, vj) = 3 for all i, j. Suppose (u1, v1) belongs to S. It strongly dominates (u2, v1), (u3, v1) and (u1, v2). If either
(u2, v2) or (u3, v2) belongs to S, leading to contradiction. Therefore they do not belong to S, a contradiction.
Similarly we get a contradiction if any (ui, vj) belongs to S. Therefore C3 □ P2 is not strong efficient.
Theorem 3.10: Km □ Pn, n, m ≥ 1 is not strong efficient.
Proof: Let V(Km) = {u1, u2, … , um} and V(Pn) = {v1, v2, … , vn}. Let V(Km □ Pn) = {(ui, vj)/1≤ i ≤ m, 1≤ j ≤ n}.
deg (ui, v1) = deg (ui, vn) = m, 1≤ i ≤ m. deg (ui, vj) = m + 1, 1 ≤ i ≤ m, 2 ≤ j ≤ n – 1. Suppose S is a strong
efficient dominating set of Km □ Pn. Suppose (ui, v2), 1 ≤ i ≤ m, belongs to S. Then (ui, v2) strongly dominates
(u1, v2), (u2, v2), …, (um, v2), (ui, v1) and (ui, v3). To strongly dominate (uk, v1), 1 ≤ k ≤ m, k i, S must contain
one of (uk, v1), 1 ≤ k ≤ m, k i. Hence (ui, v1) is strongly dominated by two elements (u k, v1) and (ui, v2) of S, a
contradiction. Similarly we get a contradiction if any (ui, vj) belongs to S. Therefore Km □ Pn is not strong
efficient.
Theorem 3.11: Km □ Cn, m ≥ 1, n ≥ 3.
Proof: Let V(Km) = {u1, u2, … , um} and V(Cn) = {v1, v2, … , vn}. Let V(Km □ Cn) = {(ui, vj)/1≤ i ≤ m, 1≤ j ≤
n}. deg(u, v) = m + 1 for all (u, v) V(Km □ Cn). Suppose S is a strong efficient dominating set of K m □ Cn.
Suppose (ui, v1), 1 ≤ i ≤ m, belongs to S. Then (ui, v1) strongly dominates (u1, v1), (u2, v1), … , (um, v1), (ui, v2)
and (ui, vn). To strongly dominate (uk, v2), 1 ≤ k ≤ m, k i, S must contain one of (uk, v2), 1 ≤ k ≤ m, k i.
Hence (ui, v2) is strongly dominated by two elements (u k, v2) and (ui, v1) of S, a contradiction. Similarly we get a
contradiction if any (ui, vj) belongs to S. Therefore K m □ Cn is not strong efficient.
Theorem 3.12:
Km □ Kn, m, n ≥ 2, is not strong efficient.
Proof: V(Km) = {u1, u2, … , um} and V(Kn) = {v1, v2, … , vn}. Let V(Km □ Kn) = {(ui, vj)/1≤ i ≤ m, 1≤ j ≤ n}.
deg(u, v) = m + n – 2 for all (u, v) V(Km □ Kn). Suppose S is a strong efficient dominating set of K m □ Kn.
Suppose (ui, v1), 1 ≤ i ≤ m, belongs to S. Then (ui, v1) strongly dominates (u1, v1), (u2, v1), … , (um, v1), (ui, v2),
(ui, v3) … (ui, vn). Suppose any (uj, vk), 1 ≤ j ≤ m, j i, 1 ≤ k ≤ n, k 1 belongs to S then (uj, v1), j 1 is
adjacent with two elements (ui, v1) and (uj, vk) of S, a contradiction. Similarly we get a contradiction if any
(ui, vj) belongs to S. Therefore Km □ Kn is not strong efficient.
Theorem 3.13:
Km □ K1,n, m ≥ 1, n ≥ 3, is not strong efficient.
Proof: Let V(Km) = {u1, u2 … um} and V(K1,n) = {v1, v2 … vn+1}. Let V(Km □ K1,n) = {(ui, vj)/1≤ i ≤ m, 1≤ j ≤
n}. For 1 ≤ i ≤ m, deg (ui, v1) = n + m – 1 = (Km □ K1,n) and deg(ui, vj) = m, 2 ≤ j ≤ n. Let S be a strong
efficient dominating set of Km □ K1,n. If any one of {(ui, v1), 1 ≤ i ≤ m} belongs to S, then it strongly uniquely
dominates (ui, v1) 1 ≤ i ≤ m, and (ui, vk), 2 ≤ k ≤ n + 1. If any one of {(uj, vk) / 1 ≤ j ≤ m} belongs to S, then
(u1, vk) is strongly dominated by two vertices (u1, v1) and (uj, vk), a contradiction. The argument is similar if any
one of {(uj, vk) / 1 ≤ j ≤ m, 1 ≤ k ≤ n+1}. Therefore Km □ K1,n is not strong efficient.
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