The document discusses restrained domination in graphs. It defines a restrained dominating set of a lict graph as a set where every vertex not in the set is adjacent to a vertex in the set as well as another vertex not in the set. It studies the exact values of the restrained domination number (the minimum cardinality of a restrained dominating set) for some standard graphs. Bounds on the restrained domination number are obtained in terms of the number of vertices and edges of the graph. Relationships between the restrained domination number and other graph parameters like diameter and domination number are also investigated.
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 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.
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.
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.
The document summarizes research characterizing graphs with specific relationships between their chromatic number (χ), domination number (γ), and complementary connected domination number (γcc). It is shown that a graph has γcc = χ = 2 if and only if it is isomorphic to a graph formed by adding two vertices to a bipartite graph. For r-regular graphs, γcc = χ = 2 if and only if the graph is isomorphic to either Kr,r or Kr+1,r+1 minus a matching.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
The document discusses claw decompositions of product graphs. It provides necessary and sufficient conditions for decomposing the Cartesian product of standard graphs into claws. Some key results are:
1. If G1 and G2 are claw decomposable, then their Cartesian product G1 x G2 is also claw decomposable.
2. The Cartesian product of complete graphs or complete bipartite graphs is claw decomposable if certain conditions on their vertex sets are met.
3. Sufficient conditions are given for the lexicographic product of a graph G with Km, Kn, or K2 x Kn to be claw decomposable when m and n satisfy certain properties.
So in summary, the document
Connected domination in block subdivision graphs of graphsAlexander Decker
This document discusses connected domination in block subdivision graphs. It begins by defining key terms such as domination, connected domination, block subdivision graphs, and connected domination number. It then presents three theorems that establish bounds on the connected domination number of block subdivision graphs in terms of the number of vertices and cut vertices of the original graph G.
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 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.
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.
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.
The document summarizes research characterizing graphs with specific relationships between their chromatic number (χ), domination number (γ), and complementary connected domination number (γcc). It is shown that a graph has γcc = χ = 2 if and only if it is isomorphic to a graph formed by adding two vertices to a bipartite graph. For r-regular graphs, γcc = χ = 2 if and only if the graph is isomorphic to either Kr,r or Kr+1,r+1 minus a matching.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
The document discusses claw decompositions of product graphs. It provides necessary and sufficient conditions for decomposing the Cartesian product of standard graphs into claws. Some key results are:
1. If G1 and G2 are claw decomposable, then their Cartesian product G1 x G2 is also claw decomposable.
2. The Cartesian product of complete graphs or complete bipartite graphs is claw decomposable if certain conditions on their vertex sets are met.
3. Sufficient conditions are given for the lexicographic product of a graph G with Km, Kn, or K2 x Kn to be claw decomposable when m and n satisfy certain properties.
So in summary, the document
Connected domination in block subdivision graphs of graphsAlexander Decker
This document discusses connected domination in block subdivision graphs. It begins by defining key terms such as domination, connected domination, block subdivision graphs, and connected domination number. It then presents three theorems that establish bounds on the connected domination number of block subdivision graphs in terms of the number of vertices and cut vertices of the original graph G.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
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.
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.
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
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.
ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONSIAEME Publication
A topological index is the graph invariant numerical descriptor calculated from a molecular graph representing a molecule. Classical Zagreb indices and the recently introduced Zagreb coindices are topological indices related to the atom – atom connectivity of the molecular structure represented by the graph G. We explore here their basic mathematical properties and present explicit formulae for these new graph invariants under several graph operations.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. It begins by defining an MST as a spanning tree (connected acyclic graph containing all vertices) with minimum total edge weight. Prim's algorithm grows a single tree by repeatedly adding the minimum weight edge connecting the growing tree to another vertex. Kruskal's algorithm grows a forest by repeatedly merging two components via the minimum weight edge connecting them. Both algorithms produce optimal MSTs by adding only "safe" edges that cannot be part of a cycle.
The Kruskal algorithm is used to find the minimum spanning tree (MST) of a connected undirected weighted graph. It works by sorting the edges in non-decreasing order of their weight, and then selecting edges one by one if they do not form a cycle with the previously selected edges. The algorithm is applied on the given graph, sorting its edges by weight and sequentially selecting edges that do not form cycles. This results in a MST with total weight of 38.
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.
IRJET- Common Fixed Point Results in Menger SpacesIRJET Journal
This document presents a common fixed point theorem for five self-maps on a complete Menger space. The theorem proves that if the maps satisfy certain conditions, including being continuous, having a compatibility property, and satisfying a contraction condition, then the maps have a common fixed point. The conditions and proof involve the use of probabilistic distances, triangular norms, Cauchy sequences, and limits in Menger spaces. The theorem generalizes prior results on common fixed points and provides a way to establish the existence of solutions to equations involving multiple operators.
sarminIJMA1-4-2015 forth paper after been publishingMustafa El-sanfaz
This document discusses the probability that a group element fixes a set and its application to generalized conjugacy class graphs. It begins with background on commutativity degree and graph theory concepts. It then reviews previous work calculating the probability for various groups and defining generalized conjugacy class graphs. The main results calculate the probability for semi-dihedral and quasi-dihedral groups as 1/2 and 1/3 respectively, and determine that the corresponding generalized conjugacy class graphs are K2 and Ke (empty graph).
On Decomposition of gr* - closed set in Topological Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
This document discusses the probability that an element of a metacyclic 2-group fixes a set. It begins by providing background information on metacyclic groups and related concepts such as commutativity degree. It then summarizes previous works that have studied the probability of a group element fixing a set. The main results presented calculate the probability for various metacyclic 2-groups of negative type with nilpotency class of at least two. In particular, it determines the probability for metacyclic 2-groups of negative type with nilpotency class of at least three. It also calculates the probability for other specific metacyclic 2-group structures.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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 presents an algorithm to find an optimal L(2,1)-labeling for triangular windmill graphs. It begins with definitions of triangular windmill graphs, L(2,1)-labelings, and the Chang-Kuo algorithm. The Chang-Kuo algorithm is then applied to obtain an L(2,1)-labeling of a triangular windmill graph W(3,n) by iteratively finding and labeling maximal 2-stable sets. The maximum label used is the labeling number λ(G).
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
“Remedies over the obstacles in implementing automation in indian infrastruct...eSAT Publishing House
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
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
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
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.
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.
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
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.
ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONSIAEME Publication
A topological index is the graph invariant numerical descriptor calculated from a molecular graph representing a molecule. Classical Zagreb indices and the recently introduced Zagreb coindices are topological indices related to the atom – atom connectivity of the molecular structure represented by the graph G. We explore here their basic mathematical properties and present explicit formulae for these new graph invariants under several graph operations.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. It begins by defining an MST as a spanning tree (connected acyclic graph containing all vertices) with minimum total edge weight. Prim's algorithm grows a single tree by repeatedly adding the minimum weight edge connecting the growing tree to another vertex. Kruskal's algorithm grows a forest by repeatedly merging two components via the minimum weight edge connecting them. Both algorithms produce optimal MSTs by adding only "safe" edges that cannot be part of a cycle.
The Kruskal algorithm is used to find the minimum spanning tree (MST) of a connected undirected weighted graph. It works by sorting the edges in non-decreasing order of their weight, and then selecting edges one by one if they do not form a cycle with the previously selected edges. The algorithm is applied on the given graph, sorting its edges by weight and sequentially selecting edges that do not form cycles. This results in a MST with total weight of 38.
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.
IRJET- Common Fixed Point Results in Menger SpacesIRJET Journal
This document presents a common fixed point theorem for five self-maps on a complete Menger space. The theorem proves that if the maps satisfy certain conditions, including being continuous, having a compatibility property, and satisfying a contraction condition, then the maps have a common fixed point. The conditions and proof involve the use of probabilistic distances, triangular norms, Cauchy sequences, and limits in Menger spaces. The theorem generalizes prior results on common fixed points and provides a way to establish the existence of solutions to equations involving multiple operators.
sarminIJMA1-4-2015 forth paper after been publishingMustafa El-sanfaz
This document discusses the probability that a group element fixes a set and its application to generalized conjugacy class graphs. It begins with background on commutativity degree and graph theory concepts. It then reviews previous work calculating the probability for various groups and defining generalized conjugacy class graphs. The main results calculate the probability for semi-dihedral and quasi-dihedral groups as 1/2 and 1/3 respectively, and determine that the corresponding generalized conjugacy class graphs are K2 and Ke (empty graph).
On Decomposition of gr* - closed set in Topological Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
This document discusses the probability that an element of a metacyclic 2-group fixes a set. It begins by providing background information on metacyclic groups and related concepts such as commutativity degree. It then summarizes previous works that have studied the probability of a group element fixing a set. The main results presented calculate the probability for various metacyclic 2-groups of negative type with nilpotency class of at least two. In particular, it determines the probability for metacyclic 2-groups of negative type with nilpotency class of at least three. It also calculates the probability for other specific metacyclic 2-group structures.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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 presents an algorithm to find an optimal L(2,1)-labeling for triangular windmill graphs. It begins with definitions of triangular windmill graphs, L(2,1)-labelings, and the Chang-Kuo algorithm. The Chang-Kuo algorithm is then applied to obtain an L(2,1)-labeling of a triangular windmill graph W(3,n) by iteratively finding and labeling maximal 2-stable sets. The maximum label used is the labeling number λ(G).
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
“Remedies over the obstacles in implementing automation in indian infrastruct...eSAT Publishing House
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
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
Online social network mining current trends and research issueseSAT Publishing House
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
Analysis of mhd non darcian boundary layer flow and heat transfer over an exp...eSAT Publishing House
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
Integration of artificial intelligence control to the unified power quality c...eSAT Publishing House
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
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
Design of wide band microstrip array antenna using direct coupled techniqueeSAT Publishing House
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.
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.
Design features of a 5 tonne day multi – stage, intermittent drainage, conti...eSAT Publishing House
This document provides details on the mechanical design of a 5 tonne per day vegetable oil solvent extraction plant. It includes calculations for the design of major components like the hopper, extractor, launder, and sumps. Dimensions and capacities are calculated based on process parameters established in a previous design. Stress analysis is performed on components like the hopper cylinder. The total volume of the designed extractor is calculated to be 2.17m3 and the estimated cost is about 3.5 million Naira.
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
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.
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.
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
The document describes the synthesis and morphology of silicon nanoparticles deposited on a silicon dioxide substrate using low pressure chemical vapor deposition with varying deposition times. Atomic force microscopy and image analysis software were used to characterize the nanoparticles and found that their height, density, and size varied with deposition time, with heights between 1-3 nm, densities from 2x1011 to 3.5x1011 particles/cm2, and sizes of 2-10 nm. The goal was to study how the morphological and electrical characteristics of the nanoparticles changed with different deposition parameters.
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
Review study on performance of seismically tested repaired shear wallseSAT Publishing House
This document summarizes research on the performance of reinforced concrete shear walls that have been repaired after damage. It begins with an introduction to shear walls and their failure modes. The literature review then discusses the behavior of original shear walls as well as different repair techniques tested by other researchers, including conventional repair with new concrete, jacketing with steel plates or concrete, and use of fiber reinforced polymers. The document focuses on evaluating the strength retention of shear walls after being repaired with various methods.
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 TechnologyIJRET : 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 various privacy preservation techniques in data mining. It summarizes classification, clustering, and association rule learning as common privacy preservation approaches. For classification, it describes decision trees, k-nearest neighbors, artificial neural networks, support vector machines, and naive Bayes models. It provides advantages and disadvantages of these techniques. The document concludes that privacy preservation techniques have emerged to allow for efficient and effective data mining while protecting sensitive data.
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 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
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.
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.
Abstract
A Roman dominating function on a graph G is a function f :V 0,1,2 satisfying the condition that every vertex uV for
which f u 0 is adjacent to at least one vertex vV for which f (v) 2 . The weight of a Roman dominating function is the
value ( ) ( )
v V
f V f v
. The Roman domination number R G of G is the minimum weight of a Roman dominating function on
G . A Roman dominating function on G is connected Roman dominating function of G if either 1 2 V V or 2 V is connected.
The connected Roman domination number RC G of G is the minimum weight of a connected Roman dominating function onG .
In this paper we establish the upper bounds, lower bounds and some equality results for RC G .
Keywords: Domination number, Roman domination number and Connected Roman domination number.
Subject Classification number: 05C69, 05C70.
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.
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.
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 presents an algorithm for calculating the number of spanning trees in chained graphs. It begins by reviewing relevant graph theory concepts like planar graphs, spanning trees, and recursive formulas for counting spanning trees using deletion/contraction and splitting methods. It then derives explicit recursions for counting spanning trees in families of graphs like wheel graphs, fan graphs, and corn graphs. The main result is a theorem providing a system of equations to calculate the number of spanning trees in a chained graph based on splitting it into components and accounting for the connecting paths. Applications to counting spanning trees in chained wheel graphs and chained corn graphs are discussed.
This document defines and discusses the complementary perfect triple connected domination number of a graph. It begins by introducing concepts like triple connected graphs and triple connected dominating sets. It then defines a complementary perfect triple connected dominating set as a triple connected dominating set where the induced subgraph on the remaining vertices has a perfect matching. The complementary perfect triple connected domination number is the minimum cardinality of such sets. The document determines this number for some standard graph classes and establishes bounds for general graphs, exploring relationships with other graph parameters.
The document introduces the concept of a restrained triple connected dominating set and restrained triple connected domination number (γrtc) of a graph. A restrained triple connected dominating set is a restrained dominating set where the induced subgraph is triple connected. γrtc is defined as the minimum cardinality of a restrained triple connected dominating set. Bounds on γrtc are provided for general graphs. Exact values of γrtc are given for certain standard graphs like cycles, complete graphs, and complete bipartite graphs. Properties of γrtc are explored, such as the relationship between a graph and its spanning subgraphs.
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
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 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 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 introduction to petal graphs, basic definitions, and a discussion of the planarity of p-petal graphs with a main theorem.
New Classes of Odd Graceful Graphs - M. E. Abdel-AalGiselleginaGloria
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ≥ 2, n ≥ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
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
The n -path graph PG (G) n of a graph G is a graph having the same vertex set as G and 2 vertices u and v in PG (G) n are adjacent if and only if there exist a path of length n between
u and v in G . In this paper we find n -path graph of some standard graphs. Bounds are given for the degree of a vertex in PG (G) n . We further characterise graphs G with PG2 (G) = G , PG2 (G) = G and PG G = Kn
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
The document summarizes research on Hamiltonian graphs. It begins by describing Sir William Hamilton's 18th century mathematical game that inspired the concept of Hamiltonian graphs. It then provides definitions and theorems related to Hamiltonian graphs, including Dirac's theorem from 1952 and Chvátal's theorem from 1972, which provide necessary conditions for a graph to be Hamiltonian based on the degrees of its vertices. The document concludes that while these theorems are useful for determining whether graphs are Hamiltonian, they are not sufficient tests on their own as some graphs satisfy the theorems but are non-Hamiltonian, or vice versa.
Similar to Restrained lict domination in graphs (20)
Hudhud cyclone caused extensive damage in Visakhapatnam, India in October 2014, especially to tree cover. This will likely impact the local environment in several ways: increased air pollution as trees absorb less; higher temperatures without tree canopy; increased erosion and landslides. It also created large amounts of waste from destroyed trees. Proper management of solid waste is needed to prevent disease spread. Suggested measures include restoring damaged plants, building fountains to reduce heat, mandating light-colored buildings, improving waste management, and educating public on health risks. Overall, changes are needed to water, land, and waste practices to rebuild the environment after the cyclone removed green cover.
Impact of flood disaster in a drought prone area – case study of alampur vill...eSAT Publishing House
1) In September-October 2009, unprecedented heavy rainfall and dam releases caused widespread flooding in Alampur village in Mahabub Nagar district, a historically drought-prone area.
2) The flood damaged or destroyed homes, buildings, infrastructure, crops, and documents. It displaced many residents and cut off the village.
3) The socioeconomic conditions and mud-based construction of homes in the village exacerbated the flood's impacts, making damage more severe and recovery more difficult.
The document summarizes the Hudhud cyclone that struck Visakhapatnam, India in October 2014. It describes the cyclone's formation, rapid intensification to winds of 175 km/h, and landfall near Visakhapatnam. The cyclone caused extensive damage estimated at over $1 billion and at least 109 deaths in India and Nepal. Infrastructure like buildings, bridges, and power lines were destroyed. Crops and fishing boats were also damaged. The document then discusses coping strategies and improvements needed to disaster management plans to better prepare for future cyclones.
Groundwater investigation using geophysical methods a case study of pydibhim...eSAT Publishing House
This document summarizes the results of a geophysical investigation using vertical electrical sounding (VES) methods at 13 locations around an industrial area in India. The VES data was interpreted to generate geo-electric sections and pseudo-sections showing subsurface resistivity variations. Three main layers were typically identified - a high resistivity topsoil, a weathered middle layer, and a basement rock. Pseudo-sections revealed relatively more weathered areas in the northwest and southwest. Resistivity sections helped identify zones of possible high groundwater potential based on low resistivity anomalies sandwiched between more resistive layers. The study concluded the electrical resistivity method was useful for understanding subsurface geology and identifying areas prospective for groundwater exploration.
Flood related disasters concerned to urban flooding in bangalore, indiaeSAT Publishing House
1. The document discusses urban flooding in Bangalore, India. It describes how factors like heavy rainfall, population growth, and improper land use have contributed to increased flooding in the city.
2. Flooding events in 2013 are analyzed in detail. A November rainfall caused runoff six times higher than the drainage capacity, inundating low-lying residential areas.
3. Impacts of urban flooding include disrupted daily life, damaged infrastructure, and decreased economic activity in affected areas. The document calls for improved flood management strategies to better mitigate urban flooding risks in Bangalore.
Enhancing post disaster recovery by optimal infrastructure capacity buildingeSAT Publishing House
This document discusses enhancing post-disaster recovery through optimal infrastructure capacity building. It presents a model to minimize the cost of meeting demand using auxiliary capacities when disaster damages infrastructure. The model uses genetic algorithms to select optimal capacity combinations. The document reviews how infrastructure provides vital services supporting recovery activities and discusses classifying infrastructure into six types. When disaster reduces infrastructure services, a gap forms between community demands and available support, hindering recovery. The proposed research aims to identify this gap and optimize capacity selection to fill it cost-effectively.
Effect of lintel and lintel band on the global performance of reinforced conc...eSAT Publishing House
This document analyzes the effect of lintels and lintel bands on the seismic performance of reinforced concrete masonry infilled frames through non-linear static pushover analysis. Four frame models are considered: a frame with a full masonry infill wall; a frame with a central opening but no lintel/band; a frame with a lintel above the opening; and a frame with a lintel band above the opening. The results show that the full infill wall model has 27% higher stiffness and 32% higher strength than the model with just an opening. Models with lintels or lintel bands have slightly higher strength and stiffness than the model with just an opening. The document concludes lintels and lintel
Wind damage to trees in the gitam university campus at visakhapatnam by cyclo...eSAT Publishing House
1) A cyclone with wind speeds of 175-200 kph caused massive damage to the green cover of Gitam University campus in Visakhapatnam, India. Thousands of trees were uprooted or damaged.
2) A study assessed different types of damage to trees from the cyclone, including defoliation, salt spray damage, damage to stems/branches, and uprooting. Certain tree species were more vulnerable than others.
3) The results of the study can help in selecting more wind-resistant tree species for future planting and reducing damage from future storms.
Wind damage to buildings, infrastrucuture and landscape elements along the be...eSAT Publishing House
1) A visual study was conducted to assess wind damage from Cyclone Hudhud along the 27km Visakha-Bheemli Beach road in Visakhapatnam, India.
2) Residential and commercial buildings suffered extensive roof damage, while glass facades on hotels and restaurants were shattered. Infrastructure like electricity poles and bus shelters were destroyed.
3) Landscape elements faced damage, including collapsed trees that damaged pavements, and debris in parks. The cyclone wiped out over half the city's green cover and caused beach erosion around protected areas.
1) The document reviews factors that influence the shear strength of reinforced concrete deep beams, including compressive strength of concrete, percentage of tension reinforcement, vertical and horizontal web reinforcement, aggregate interlock, shear span-to-depth ratio, loading distribution, side cover, and beam depth.
2) It finds that compressive strength of concrete, tension reinforcement percentage, and web reinforcement all increase shear strength, while shear strength decreases as shear span-to-depth ratio increases.
3) The distribution and amount of vertical and horizontal web reinforcement also affects shear strength, but closely spaced stirrups do not necessarily enhance capacity or performance.
Role of voluntary teams of professional engineers in dissater management – ex...eSAT Publishing House
1) A team of 17 professional engineers from various disciplines called the "Griha Seva" team volunteered after the 2001 Gujarat earthquake to provide technical assistance.
2) The team conducted site visits, assessments, testing and recommended retrofitting strategies for damaged structures in Bhuj and Ahmedabad. They were able to fully assess and retrofit 20 buildings in Ahmedabad.
3) Factors observed that exacerbated the earthquake's impacts included unplanned construction, non-engineered buildings, improper prior retrofitting, and defective materials and workmanship. The professional engineers' technical expertise was crucial for effective post-disaster management.
This document discusses risk analysis and environmental hazard management. It begins by defining risk, hazard, and toxicity. It then outlines the steps involved in hazard identification, including HAZID, HAZOP, and HAZAN. The document presents a case study of a hypothetical gas collecting station, identifying potential accidents and hazards. It discusses quantitative and qualitative approaches to risk analysis, including calculating a fire and explosion index. The document concludes by discussing hazard management strategies like preventative measures, control measures, fire protection, relief operations, and the importance of training personnel on safety.
Monitoring and assessment of air quality with reference to dust particles (pm...eSAT Publishing House
This document summarizes a study on monitoring and assessing air quality with respect to dust particles (PM10 and PM2.5) in the urban environment of Visakhapatnam, India. Sampling was conducted in residential, commercial, and industrial areas from October 2013 to August 2014. The average PM2.5 and PM10 concentrations were within limits in residential areas but moderate to high in commercial and industrial areas. Exceedance factor levels indicated moderate pollution for residential areas and moderate to high pollution for commercial and industrial areas. There is a need for management measures like improved public transport and green spaces to combat particulate air pollution in the study areas.
Low cost wireless sensor networks and smartphone applications for disaster ma...eSAT Publishing House
This document describes a low-cost wireless sensor network and smartphone application system for disaster management. The system uses an Arduino-based wireless sensor network comprising nodes with various sensors to monitor the environment. The sensor data is transmitted to a central gateway and then to the cloud for analysis. A smartphone app connected to the cloud can detect disasters from the sensor data and send real-time alerts to users to help with early evacuation. The system aims to provide low-cost localized disaster detection and warnings to improve safety.
Coastal zones – seismic vulnerability an analysis from east coast of indiaeSAT Publishing House
This document summarizes an analysis of seismic vulnerability along the east coast of India. It discusses the geotectonic setting of the region as a passive continental margin and reports some moderate seismic activity from offshore in recent decades. While seismic stability cannot be assumed given events like the 2004 tsunami, no major earthquakes have been recorded along this coast historically. The document calls for further study of active faults, neotectonics, and implementation of improved seismic building codes to mitigate vulnerability.
Can fracture mechanics predict damage due disaster of structureseSAT Publishing House
This document discusses how fracture mechanics can be used to better predict damage and failure of structures. It notes that current design codes are based on small-scale laboratory tests and do not account for size effects, which can lead to more brittle failures in larger structures. The document outlines how fracture mechanics considers factors like size effect, ductility, and minimum reinforcement that influence the strength and failure behavior of structures. It provides examples of how fracture mechanics has been applied to problems like evaluating shear strength in deep beams and investigating a failure of an oil platform structure. The document argues that fracture mechanics provides a more scientific basis for structural design compared to existing empirical code provisions.
This document discusses the assessment of seismic susceptibility of reinforced concrete (RC) buildings. It begins with an introduction to earthquakes and the importance of vulnerability assessment in mitigating earthquake risks and losses. It then describes modeling the nonlinear behavior of RC building elements and performing pushover analysis to evaluate building performance. The document outlines modeling RC frames and developing moment-curvature relationships. It also summarizes the results of pushover analyses on sample 2D and 3D RC frames with and without shear walls. The conclusions emphasize that pushover analysis effectively assesses building properties but has limitations, and that capacity spectrum method provides appropriate results for evaluating building response and retrofitting impact.
A geophysical insight of earthquake occurred on 21 st may 2014 off paradip, b...eSAT Publishing House
1) A 6.0 magnitude earthquake occurred off the coast of Paradip, Odisha in the Bay of Bengal on May 21, 2014 at a depth of around 40 km.
2) Analysis of magnetic and bathymetric data from the area revealed the presence of major lineaments in NW-SE and NE-SW directions that may be responsible for seismic activity through stress release.
3) Movements along growth faults at the margins of large Bengal channels, due to large sediment loads, could also contribute to seismic events by triggering movements along the faults.
Effect of hudhud cyclone on the development of visakhapatnam as smart and gre...eSAT Publishing House
This document discusses the effects of Cyclone Hudhud on the development of Visakhapatnam as a smart and green city through a case study and preliminary surveys. The surveys found that 31% of participants had experienced cyclones, 9% floods, and 59% landslides previously in Visakhapatnam. Awareness of disaster alarming systems increased from 14% before the 2004 tsunami to 85% during Cyclone Hudhud, while awareness of disaster management systems increased from 50% before the tsunami to 94% during Hudhud. The surveys indicate that initiatives after the tsunami improved awareness and preparedness. Developing Visakhapatnam as a smart, green city should consider governance
This document summarizes a product called H-HUTS that aims to provide sustainable disaster recovery housing. H-HUTS uses cardboard and origami-inspired modular construction to quickly provide temporary shelters after disasters. The cardboard shelters are designed to be upgraded into permanent homes over time using low-cost materials like shotcrete or concrete. The modular design allows for flexibility and community participation in construction. Testing has shown the shelters can withstand wind loads and provide adequate insulation. The goal of H-HUTS is to not just provide initial shelter but help disaster-affected communities regain permanency in housing and recover in the long-term.
Mechatronics is a multidisciplinary field that refers to the skill sets needed in the contemporary, advanced automated manufacturing industry. At the intersection of mechanics, electronics, and computing, mechatronics specialists create simpler, smarter systems. Mechatronics is an essential foundation for the expected growth in automation and manufacturing.
Mechatronics deals with robotics, control systems, and electro-mechanical systems.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Software Engineering and Project Management - Software Testing + Agile Method...Prakhyath Rai
Software Testing: A Strategic Approach to Software Testing, Strategic Issues, Test Strategies for Conventional Software, Test Strategies for Object -Oriented Software, Validation Testing, System Testing, The Art of Debugging.
Agile Methodology: Before Agile – Waterfall, Agile Development.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
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RESTRAINED LICT DOMINATION IN GRAPHS
M.H. Muddebihal1
, Kalshetti Swati.M2
1
Professor and Chairman, Department of P.G. Studies and Research in Mathematics Gulbarga University, Gulbarga,
Karnataka, India.
2
Assistant Professor, Department of Mathematics, Godutai Engineering College for Women, Gulbarga, Karnataka,
India.
Abstract
A set rD V n G is a restrained dominating set of n G , where every vertex in rD)]G(n[V is adjacent to a vertex
in rD as well as another vertex in )]G(n[V rD .
The restrained domination number of lict graph n G , denoted by rn G , is the minimum cardinality of a restrained
dominating set of n G . In this paper, we study its exact values for some standard graphs we obtained. Also its relation with
other parameters is investigated.
Subject classification number: AMS-O5C69, 05C70
Keywords: Lict Graph / Line Graph / Restrained domination / Dominating set / Edge domination
--------------------------------------------------------------------***------------------------------------------------------------------
1. INTRODUCTION
In this paper, all the graphs considered here are simple,
finite, non-trivial, undirected and connected. As usual p
and q denote, the number of vertices and edges of a graph
G . In this paper, for any undefined terms or notations can
be found in Harary [4].
As usual, the maximum degree of a vertex in G is denoted
by G .
The degree of an edge e uv of G is defined as
deg deg deg 2e u v and GG is the
minimum(maximum) degree among the edges of G .
For any real number x , x denotes the smallest integer
not less than x and x denotes the greatest integer not
greater than x . The complement G of a graph G has V as
its vertex set, but two vertices are adjacent in G if they are
not adjacent in G .
A vertex (edge) cover in a graph G is a set of vertices that
covers all the edges (vertices) of G . The vertex (edge)
covering number 0 G ( 1 G ) is a minimum
cardinality of a vertex (edge) cover in G . The vertex (edge)
independence number GG 10 is the maximum
cardinality of independent set of vertices (edges) in G .
The greatest distance between any two vertices of a
connected graph G is called the diameter of G and is
denoted by diam G .
We begin by recalling some standard definition from
domination theory.
A set D of a graph ,G V E 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 minimal dominating set in G . The study of
domination in graphs was begun by Ore [7] and Berge [1].
A set D V L G is said to be dominating set of
L G if every vertex not in D is adjacent to a vertex in
D .The domination number of G is denoted by
L G is the minimum cardinality of dominating set in
G .
A set F of edges in a graph G is called an edge
dominating set of G if every edge in E F where E is
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the set of edges in G is adjacent to atleast one edge in F .
The edge domination number '
G of a graph G is the
minimum cardinality of an edge dominating set of G . The
concept of edge domination number in graphs were studied
by Gupta [3] and S.Mitchell and S.T.Hedetineim [6].
Analogously, we define restained domination number in lict
graph as follows.
A dominating set rD of lict graph is a restained dominating
set, if every vertex not in rD is adjacent in rD and to a
vertex in rV D .The restrained domination number of lict
graph n G ,denoted by rn G is the minimum
cardinality of a restrained dominating set of n G .The
concept of restrained domination in graphs was introduced
by Domke et al [2].
In this paper, many bounds on rn G were obtained and
expressed in terms of vertices, edges of G but not the
elements of n G . Also we establish restained domination
number of a lict graph n G and express the results with
other different domination parameters of G .
2. RESULTS
We need the following Theorems to establish our further
results.
Theorem A [5]: For any connected ,p q graph G ,
1
q
G
G
.
Theorem B [5]: If G is a graph with no isolated vertex,
then G q G .
Initially we begin with restrained domination number of lict
graph of some standard graphs, which are straight forward
in the following Theorem.
2.1 Theorem 1:
(i) For any cycle pC with 3p vertices,
2 .
3
rn p
p
C p
(ii) For any path pP with 2p vertices,
2 1rn nP k
2rn nP k , When 2,3,4,5,........n …..
Then 1,2,3,4,...........k ..
(iii) For any star 1, pK with 2p vertices,
1, 1.rn pK
(iv) For any wheel pW with 4p vertices,
3
1
3
rn p
p
W
.
(v) For any complete bipartite graph 1 2,p pK with
1 2, 2p p vertices,
1 2, 1 2min ,rn p pK p p .
(vi) For any complete graph pK with 3p vertices ,
prn K =
2
p
.
In the following Theorem, we establish the upper bound for
rn T in term of vertices of the G .
2.2 Theorem 2
For any tree T with 2p vertices and m end vertices
rn T p m . Equality holds if 1, pT K with
2p vertices.
Proof: If 3diam G , then the result is obvious. Let
3diam T and 1 1 2 3, , ,........., nV v v v v be the set
of all end vertices in T with 1V m .Further
qe..,,.........e,e,eE 321 and
ic...,,.........c,c,cC 321 be the set of edges and
cutvertices in G . In Gn , GCGEGnV
and in G ie incicident with jc , ij 1 forms a
complete induced subgraph as a block in Gn , such that
the number of blocks in CGn . Let
GEe.....,,.........e,e,e j 321 which are non end
edges of G forms cutvertices
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jc.....,,.........c,c,cGC 321 in Gn . Let
CC 1 be a restrained dominating set in Gn , such that
GC rn1 . For any non trivial tree qp and
mpC 1 which gives mpTrn .
Further equality holds if p,KT 1 then 11 pp, KKn
and .mpK p,rn 1
The following corollaries are immediate from the above
Theorem.
Corollory 1: For any connected ,p q graph G ,
0 0rn G G G G . Equality holds if
G is isomorphic to 4C or 5C .
Corollory 2: For any connected ,p q graph G ,
1 1rn G G G G . Equality holds if G
is isomorphic to 4C or 5C .
2.3 Theorem 3
For any connected ,p q graph G with 2p vertices ,
2
rn
p
G
. Equality holds if G is 4C or 5C or 8C
or pK if p is even.
Proof: Let 1 2 3, , ,............., nE e e e e be the edge set of
G such that V n G E G C G , by definition
of lict graph where C G is the set of cutvertices in G .
Let 1 2 3, , ,............,r nD v v v v V n G be the
restrained dominating set of n G . Suppose if
2rV n G D , then rV n G D contains
atleast two vertices which gives
2 2
rn
p p
G
.
For the equality,
i) If G is isomorphic to 4C or 5C or 8C .For any cycle pC
with 3p vertices p pn C C ,which gives
2
r
p
D
.Therefore 2
rn p
p
C
.
ii) If G is isomorphic to pK ,where is p even, , then by
Theorem 1 ,
2
p
K prn .
In the following Theorem, we obtain the relation between
rn G and diameter of G .
2.4 Theorem 4
For any connected ,p q graph G ,
1
.
3
rn
diam G
G
Proof: Let rD be a restrained dominating set of n G
such that r rnD G . Consider an arbitrary path of
length which is a diam G . This diameteral path induces
atmost three edges from the induced subgraph vN for
each rv D . Furthermore since rD is rn -set, the
diametral path includes atmost 1rn G edges joining
the neighborhood of the vertices of rD .
Hence 2 rndiam G G 1rn G .
3 1.rndiam G G
Hence the result follows.
The following theorem relates domination number of G
and restrained domination number of n G .
2.5 Theorem 5
For any connected ,p q graph G with 3p vertices,
rn G P
G . Equality holds if 4G C or 5C .
Proof: Let 1 2 3, , ,.........., nD u u u u be a minimal
dominating set of n G such that D G . Further
let 1 1 2 3, , ,......... nF e e e e be the set of all edges which
are incident to the vertices of D and 2 1F E G F .
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Let 1 2 3, , ,............, nC c c c c be the cutvertex set of G
. By definition of lict graph ,
V n G E G C G and 1F V n G .Let
1 1 2 3, , ,....... kI e e e e ;1 k i ,where 1 1I F and
2 2I F .Since each induced subgraph which is complete in
n G may contain atleast one vertex of either 1F or 2F .
Then 1 2I I forms a minimal restrained dominating set
in n G such that 1 2 r rnI I D G .Clearly
1 2D I I p .Thus it follows that
rnG G p .
For equality,
If pG C for p 4 or 5 , then by definition of Lict
graph p pn C C .Then in this case
2
r
p
D D .
Clearly it follows that rn G G p .
In [5] , they related )G(' with line domination of G . In
the following theorem we establish our result with edge
dominationof G .
2.6 Theorem 6
For any non-trivial connected ,p q graph G ,
.GGrn
Proof: Let n, e,........,e,eeE 321 be the edge set of G
and nc,......,c,c,cC 321 be the set of cutvertices in
G , V n G E G C G . Let
1 2 3, , ,..........., nF e e e e ;
ie , where 1 i n be the minimal edge dominating set
of G , such that F G .
Since GnVGE every edge ie F ; ie ;
1 i n forms a dominating set in n G .Suppose
1F E G F V n G , we consider
1 1 2 3, , ,..............., nI e e e e ;
1 k i , where 1I F and 2 1I F . Since each
induced subgraph which is complete in n G may contain
atleast one vertex of either F or 1F . Then 1 2I I forms
a minimal restrained dominating set in n G .Clearly it
follows that 1 2F I I in n G . Hence
rnG G .
In the next Theorem, we obtain the relation between
domination number of G and restrained domination
number of n G in terms of vertices and diameter of G .
2.7 Theorem 7
For any connected ,p q graph G with 2p vertices,
rn G p
G diam G .
Proof: Let 1 2 3, , ,......., nV v v v v be the set of vertices
in G . Suppose there exists two vertices ,u v V G such
that dist ,u v = diam G .Let
pv,........,v,v,vD 321 ; 1 p n be a minimal
dominating set in n G . Now we consider
1 2 3, , ,......., nF e e e e ; F E G and
ie V n G ,1 i n in n G .The V n G =
E G C G ,where C G is the set of cutvertices in
G .Suppose 1 1,F C are the subsets of F and C .Then a
set 1 1J V n G F C such that J has no
isolates .Clearly 1 1 rnF C G .Let ,u v V G ,
,d u v diam G , then
1 1F C diam G p D .Hence
rn G diam G p G which implies
rn G p G diam G .
The following Theorem, relates restrained domination
number of n G in terms of vertices and G .
2.8 Theorem 8
For any connected ,p q graph G ,
rn G p G .Equality holds if G is 3P or pC
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3 5p or 1, pK 2p or 1 2,p pK with
1 2, 2p p .
Proof: We consider the following cases.
Case 1: Suppose G is tree. Then
1 2 3, , ,........, nE G e e e e be the set of all end edges
and 1 2 3, , ,.........., iE G e e e e ; ie E G be the
set of all non-end edges. Let
nr u.........,,.........u,u,uD 321 be the vertex set of
n G which corresponds to the set F and is a minimal
edge dominating set of n G where GnVF =
GCGE , C G a set of cutvertices . Further let
1I V n G F and 2 1I F , where
1F E G F V n G . Then 1 2I I forms a
minimal restrained dominating set of n G , where
1 2,I I E G V n G . Since it is clear that
rn G p and 1p q .Let there exists a vertex
iv G and also by Theorem 3,
2
rn
p
, which gives
the result of minimal restrained dominating set of n G
such that rD p G and is rn G P G
.
Case 2: Suppose G is not a tree , again we consider the
following subcases of case 2.
Subcase 2.1: Assume G is a cycle pC 3p ,Since for
any cycle pC with 3p vertices 2G and by
Theorem 1, 2
3
rn p
p
C p
p G .
Subcase 2.2: Assume G is a cyclic graph .Then there
exists a cycle or block in G which contains cycles. Let
1 2 3, , ,........., nF e e e e be the edge dominating set in
G , such that atleast one of ie G ; 1 i n , where
G is a block or a cycle in G . In n G the set F gives a
minimal dominating set and let
1 1 2 3, , ,.........., nI v v v v be the set of vertices in
n G and 1I E G V n G F and let
2 1I F where 1F E G F .Then 1 2I I is a
minimal restrained dominating set of n G , which gives
1 2I I r rnD G . Suppose v G and
atleast two edges which are incident to v are the element of
F which gives p G .So one can easily verify that
rnp G G .
For equality,
i) If 3G P , then 3 3n P C and 1rD ,which gives
rn G p G .
ii) If pG C for 3 5p ,since for any cycle
p pn C C and 2 . Then rD = p G , which
gives rn pC p G .
iii) If 1, pG K for 2p vertices , then 1, p pn K K .
For any star 1p and
1p q and also 1rD ,which gives
1,rn pK p G .
iv) If 1 2,p pG K with 1 2, 2p p vertices , then in this
case 1 2max ,p p and also 1 2p p V G .
Since by Theorem 1,
1 2 1 2min , max ,rD p p V G p p , which
gives 1 2, 1 2min ,rn p pK p p p G .
The following Theorem gives the relation between the
restrained domination number of ( )n G and vertex covering
number of G .
2.9 Theorem 9
For any connected ,p q graphG with 2p vertices,
rn G 0 G
Proof: Let 1 2 3, , ,.............., mB v v v v V G be
the minimum number of vertices which covers all the edges
, such that 0B G and
1 1 2 3, , ,............., kE e e e e E G , such that
iv B ;1 i n is incident with ie , for 1 i k . We
consider the following cases.
Case 1: Suppose for any two vertices 1 2,v v B and
1 2v N v .Then an edge e incident with 1v and 2v
covers all edges incident with 1v and 2v . Hence e belongs
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to rn - set of G . Further for any vertex iv B covering
the edge 1i
e E incident with a cutvertex cv of G , ie
belongs to the rn set of G . Thus
.GBGrn 0
Case 2: Suppose for any two vertices 1 2,v v B and
2v N v .Then 1 2 1,e e E covers all the edges incident
with 1v and 2v . Since B consist of the vertices which
covers the edges that are incident all the cutvertices of G ,
the corresponding edges in 1E covers the cutvertices of G .
Thus 0rn G B G .
Next we obtain a bound of restrained lict domination
number in terms of number of edges and maximum edge
degree of G .
2.10 Theorem 10
For any connected ,p q graph G with 3p ,
rn G q G . Proof: We consider the following
cases.
Case 1 : Suppose G is non-seperable. Using Theorem 6
and Theorem B , the result follows.
Case 2 : Suppose G is seperable. Let e be an edge with
degree and let M be a set of edges adjacent to e in G
.
Then E G M covers all the edges and all the
cutvertices of G .But some of the ie s E G M , for
1 i n forms a minimal restrained dominating set in
n G .
Then rn G E G M , which gives
rn G q G .
The following Theorem relates the domination number of
L G and and restrained domination number of n G .
2.11 Theorem 11
For any connected graph G with 2p vertices,
rn G q L G
Proof: Let 1 2 3, , ,........., nE e e e e be the edge set of
G and 1 2 3, , ,........., nC c c c c be the cutvertex set of
G , then V n G E G C G and
V L G E G ,by definition. Suppose
1 2 3, , ,.........., nH u u u u V L G be the set of
vertices of degree deg 2,1iu i n .Then D H
forms a minimal dominating set of L G such that
D L G . Further let
1 2 3, , ,........, iH u u u u ;1 i n , where H H
.Then H D forms a minimal restrained dominating set
in n G . Since V L G E G q and also
V L G V n G .Clearly it follows that
D H D q . Thus rn G L G q .
To prove our further result, we give the following two
observations.
Observation 1: For connected ,p q graph G ,
2rn G q .
Proof: Suppose rD is a restrained dominating set of
n G .Then by definition of restrained domination
2V n G . Further by definition of n G ,
2rnq G . Clearly it follows that 2rn G q .
Observation 2: Suppose rD be any restrained dominating
set of n G , such that r rnD G .Then
deg
i r
r i
v D
V n G D v
.
Proof: Since every vertex in rV n G D is adjacent
to atleast one vertex in rV n G D contributes atleast
one to the sum of the degrees of the vertices of rD . Hence
the proof.
2.12Theorem 12
For any connected ,p q graph G ,
1
rn
q
G q G
G
.
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Proof: Let e E G , now without loss of generality , by
definition of lict graph , e u V n G and let rD be
the restrained dominating set of n G such that
r rnD G . If 2G ,then by observation 1,
2 .rn G q q G If 2G , then for
any edge f N e and by definition of n G ,
f w N u , rD V n G N u w .
Then 1 1rn G q G q G .
Now for the lower bound we have by observation 2 and the
fact that any edge e E G and dege G , we
have
deg .
r
rn rn
v D
q G V n G n G v G G
.
Therefore
1
rn
q
G
G
.
2.13Theorem 13
For any non-trivial connected ,p q graph G ,
1
rn
q
G
G
.
Proof: Using Theorem 6 and Theorem A, the result follows.
Finally we obtain the Nordhous – Gaddum type result.
2.14 Theorem 14
Let G be a connected ,p q graph such that both G and
G are connected, then
i) .
2
rn rn
p
G G
ii) 3
. .
2
rn rn
p
G G
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