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 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.
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.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document introduces and defines the concept of neighborhood triple connected domination number (ntc) of a graph. A neighborhood triple connected dominating set of a graph G is a dominating set where the induced subgraph of the open neighborhood of the set is triple connected. The ntc of G is the minimum cardinality of such a set. The document provides the ntc values for some standard graphs like complete graphs and wheels. It also gives ntc values for specific graphs like the diamond, fan, and Moser spindle graphs. Real-life applications of ntc sets are discussed. Properties of ntc sets are observed and examples are given.
The document discusses connectivity in graphs. It defines edge connectivity and vertex connectivity as numerical parameters that measure how connected a graph is. Edge connectivity is the minimum number of edges that need to be removed to disconnect the graph. Vertex connectivity is defined similarly for vertices. It provides examples and discusses properties like cut sets, bridges, and the relationship between these concepts and connectivity values. Menger's theorem relating the size of the minimum cut to the maximum number of disjoint paths is also covered.
International Journal of Computational Engineering Research(IJCER)ijceronline
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.
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.
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.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document introduces and defines the concept of neighborhood triple connected domination number (ntc) of a graph. A neighborhood triple connected dominating set of a graph G is a dominating set where the induced subgraph of the open neighborhood of the set is triple connected. The ntc of G is the minimum cardinality of such a set. The document provides the ntc values for some standard graphs like complete graphs and wheels. It also gives ntc values for specific graphs like the diamond, fan, and Moser spindle graphs. Real-life applications of ntc sets are discussed. Properties of ntc sets are observed and examples are given.
The document discusses connectivity in graphs. It defines edge connectivity and vertex connectivity as numerical parameters that measure how connected a graph is. Edge connectivity is the minimum number of edges that need to be removed to disconnect the graph. Vertex connectivity is defined similarly for vertices. It provides examples and discusses properties like cut sets, bridges, and the relationship between these concepts and connectivity values. Menger's theorem relating the size of the minimum cut to the maximum number of disjoint paths is also covered.
International Journal of Computational Engineering Research(IJCER)ijceronline
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.
This document defines and provides examples of key concepts in graph theory, including subgraphs, walks, paths, cycles, connectivity, and k-connectivity. It defines subgraphs, spanning subgraphs, trivial subgraphs, and induced subgraphs. It defines walks, paths, and cycles. It defines connectivity and connectivity in graphs, articulation vertices, bridges, and distance in connected graphs. It defines k-connectivity and cut vertices. It provides examples of separating sets, edge connectivity, edge cuts, and blocks.
This document provides an overview of graphs and graph algorithms. It defines graphs, directed and undirected graphs, and graph terminology like vertices, edges, paths, cycles, connected components, and degrees. It describes different graph representations like adjacency matrices and adjacency lists. It also explains graph traversal algorithms like depth-first search and breadth-first search. Finally, it covers graph algorithms for finding minimum spanning trees, shortest paths, and transitive closure.
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.
The document discusses various concepts related to connectivity in graphs. It defines what makes a graph connected versus disconnected. Key points include: a connected graph has a path between all vertex pairs; removing vertices or edges can disconnect a graph; connected components are maximal connected subgraphs; cut vertices and edges disconnect the graph when removed. Graph isomorphism is also discussed, where two graphs are isomorphic if their adjacency matrices are identical.
A study on connectivity in graph theory june 18 123easwathymaths
This document provides an introduction to connectivity of graphs. It begins with definitions of terms like bridges, cut vertices, connectivity, and edge connectivity. It then presents several theorems about when edges are bridges and vertices are cut vertices. It proves properties of trees related to cut vertices. The document establishes relationships between vertex and edge connectivity. It introduces the concepts of k-connectivity and discusses properties of complete graphs and trees in relation to connectivity.
A study on connectivity in graph theory june 18 pdfaswathymaths
This document provides an introduction and two chapters on connectivity in graphs. The introduction discusses the history and applications of graph theory. Chapter 1 defines key concepts related to connectivity such as bridges, cut vertices, and k-connectivity. It presents theorems characterizing when an edge is a bridge and when a graph is a tree. Chapter 2 discusses applications of connectivity in graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
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.
The document discusses graph theory and provides definitions and examples of various graph concepts. It defines what a graph is consisting of vertices and edges. It also defines different types of graphs such as simple graphs, multigraphs, digraphs and provides examples. It discusses graph terminology, models, degree of graphs, handshaking lemma, special graphs and applications. It also provides explanations of planar graphs, Euler's formula and graph coloring.
This document provides an introduction to graph theory concepts. It defines graphs as mathematical objects consisting of nodes and edges. Both directed and undirected graphs are discussed. Key graph properties like paths, cycles, degrees, and connectivity are defined. Classic graph problems introduced include Eulerian circuits, Hamiltonian circuits, spanning trees, and graph coloring. Graph theory is a fundamental area of mathematics with applications in artificial intelligence.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
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.
Graph theory concepts complex networks presents-rouhollah nabatinabati
This document provides an introduction to network and social network analysis theory, including basic concepts of graph theory and network structures. It defines what a network and graph are, explains what network theory techniques are used for, and gives examples of real-world networks that can be represented as graphs. It also summarizes key graph theory concepts such as nodes, edges, walks, paths, cycles, connectedness, degree, and centrality measures.
This document provides an overview of graph theory concepts including:
- The basics of graphs including definitions of vertices, edges, paths, cycles, and graph representations like adjacency matrices.
- Minimum spanning tree algorithms like Kruskal's and Prim's which find a spanning tree with minimum total edge weight.
- Graph coloring problems and their applications to scheduling problems.
- Other graph concepts covered include degree, Eulerian paths, planar graphs and graph isomorphism.
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.
A graph is planar if it can be drawn on a plane without edge crossings. The complete graphs K5 and K3,3 are non-planar as they contain subgraphs that cannot be drawn without crossings. The Euler formula relates the number of vertices, edges, and faces in a planar graph as e - n + f = 2. Planarity can be tested using Kuratowski's theorem which states that a graph is planar unless it contains K5 or K3,3 as a subgraph.
Graph theory with algorithms and its applicationsSpringer
The document discusses subgraphs, paths, and connected graphs in graph theory. It defines key terms like subgraph, spanning subgraph, walk, trail, path, connected graph, disconnected graph, and components. It also covers operations on graphs like union, intersection, ring sum, deletion, and fusion. Graphs can be decomposed or have induced subgraphs. The document proves theorems about connected graphs and provides definitions for cycles.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
El documento habla sobre la transición a proyectos ágiles en una organización. Explica los beneficios que promete el enfoque ágil como mejorar la gestión de proyectos, entregar valor al cliente más rápido y mejorar la calidad. También discute los desafíos de implementar prácticas ágiles como cambios en roles y estructuras y posibles resistencias al cambio. Recomienda empezar con pequeños cambios de forma incremental para lograr una transición exitosa.
Este documento presenta recetas para albóndigas con hongos y arroz, y fresas sumergidas en natas. La receta de albóndigas incluye cocinar una mezcla de cebolla y ajo con aceite de oliva, luego agregar hongos, albóndigas y cerveza y cocinar durante 30-40 minutos, y servir con arroz hervido por separado. La receta de fresas consiste en cortar fresas en trozos y mezclarlas con natas y azúcar para servirlas frescas
This document defines and provides examples of key concepts in graph theory, including subgraphs, walks, paths, cycles, connectivity, and k-connectivity. It defines subgraphs, spanning subgraphs, trivial subgraphs, and induced subgraphs. It defines walks, paths, and cycles. It defines connectivity and connectivity in graphs, articulation vertices, bridges, and distance in connected graphs. It defines k-connectivity and cut vertices. It provides examples of separating sets, edge connectivity, edge cuts, and blocks.
This document provides an overview of graphs and graph algorithms. It defines graphs, directed and undirected graphs, and graph terminology like vertices, edges, paths, cycles, connected components, and degrees. It describes different graph representations like adjacency matrices and adjacency lists. It also explains graph traversal algorithms like depth-first search and breadth-first search. Finally, it covers graph algorithms for finding minimum spanning trees, shortest paths, and transitive closure.
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.
The document discusses various concepts related to connectivity in graphs. It defines what makes a graph connected versus disconnected. Key points include: a connected graph has a path between all vertex pairs; removing vertices or edges can disconnect a graph; connected components are maximal connected subgraphs; cut vertices and edges disconnect the graph when removed. Graph isomorphism is also discussed, where two graphs are isomorphic if their adjacency matrices are identical.
A study on connectivity in graph theory june 18 123easwathymaths
This document provides an introduction to connectivity of graphs. It begins with definitions of terms like bridges, cut vertices, connectivity, and edge connectivity. It then presents several theorems about when edges are bridges and vertices are cut vertices. It proves properties of trees related to cut vertices. The document establishes relationships between vertex and edge connectivity. It introduces the concepts of k-connectivity and discusses properties of complete graphs and trees in relation to connectivity.
A study on connectivity in graph theory june 18 pdfaswathymaths
This document provides an introduction and two chapters on connectivity in graphs. The introduction discusses the history and applications of graph theory. Chapter 1 defines key concepts related to connectivity such as bridges, cut vertices, and k-connectivity. It presents theorems characterizing when an edge is a bridge and when a graph is a tree. Chapter 2 discusses applications of connectivity in graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
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.
The document discusses graph theory and provides definitions and examples of various graph concepts. It defines what a graph is consisting of vertices and edges. It also defines different types of graphs such as simple graphs, multigraphs, digraphs and provides examples. It discusses graph terminology, models, degree of graphs, handshaking lemma, special graphs and applications. It also provides explanations of planar graphs, Euler's formula and graph coloring.
This document provides an introduction to graph theory concepts. It defines graphs as mathematical objects consisting of nodes and edges. Both directed and undirected graphs are discussed. Key graph properties like paths, cycles, degrees, and connectivity are defined. Classic graph problems introduced include Eulerian circuits, Hamiltonian circuits, spanning trees, and graph coloring. Graph theory is a fundamental area of mathematics with applications in artificial intelligence.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
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.
Graph theory concepts complex networks presents-rouhollah nabatinabati
This document provides an introduction to network and social network analysis theory, including basic concepts of graph theory and network structures. It defines what a network and graph are, explains what network theory techniques are used for, and gives examples of real-world networks that can be represented as graphs. It also summarizes key graph theory concepts such as nodes, edges, walks, paths, cycles, connectedness, degree, and centrality measures.
This document provides an overview of graph theory concepts including:
- The basics of graphs including definitions of vertices, edges, paths, cycles, and graph representations like adjacency matrices.
- Minimum spanning tree algorithms like Kruskal's and Prim's which find a spanning tree with minimum total edge weight.
- Graph coloring problems and their applications to scheduling problems.
- Other graph concepts covered include degree, Eulerian paths, planar graphs and graph isomorphism.
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.
A graph is planar if it can be drawn on a plane without edge crossings. The complete graphs K5 and K3,3 are non-planar as they contain subgraphs that cannot be drawn without crossings. The Euler formula relates the number of vertices, edges, and faces in a planar graph as e - n + f = 2. Planarity can be tested using Kuratowski's theorem which states that a graph is planar unless it contains K5 or K3,3 as a subgraph.
Graph theory with algorithms and its applicationsSpringer
The document discusses subgraphs, paths, and connected graphs in graph theory. It defines key terms like subgraph, spanning subgraph, walk, trail, path, connected graph, disconnected graph, and components. It also covers operations on graphs like union, intersection, ring sum, deletion, and fusion. Graphs can be decomposed or have induced subgraphs. The document proves theorems about connected graphs and provides definitions for cycles.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
El documento habla sobre la transición a proyectos ágiles en una organización. Explica los beneficios que promete el enfoque ágil como mejorar la gestión de proyectos, entregar valor al cliente más rápido y mejorar la calidad. También discute los desafíos de implementar prácticas ágiles como cambios en roles y estructuras y posibles resistencias al cambio. Recomienda empezar con pequeños cambios de forma incremental para lograr una transición exitosa.
Este documento presenta recetas para albóndigas con hongos y arroz, y fresas sumergidas en natas. La receta de albóndigas incluye cocinar una mezcla de cebolla y ajo con aceite de oliva, luego agregar hongos, albóndigas y cerveza y cocinar durante 30-40 minutos, y servir con arroz hervido por separado. La receta de fresas consiste en cortar fresas en trozos y mezclarlas con natas y azúcar para servirlas frescas
El documento describe un programa en C que suma y multiplica dos números decimales introducidos por el usuario utilizando funciones. El programa define funciones para la suma y multiplicación que toman los números como parámetros y devuelven el resultado. Luego solicita los números al usuario, llama a las funciones y muestra los resultados de la suma y multiplicación.
El documento analiza la evolución del uso de Internet por parte de los gobiernos y administraciones vascas. Explica que están mejorando su presencia en la red y empiezan a entender el valor de trabajar en red. También destaca los retos de implementar e-administración, e-democracia y e-gobernanza para ser más receptivos a los ciudadanos y fomentar su participación.
Paco experimentaba dolor en la muñeca después del trabajo en la oficina, pero Encarni ya no tenía molestias después de comprar una alfombrilla ergonómica. La ergonomía se refiere al diseño de productos, ambientes y trabajos para adaptarse a las personas y mejorar la calidad, eficiencia y salud del trabajador. Entornos y herramientas de trabajo mal diseñados pueden causar lesiones y enfermedades a largo plazo.
An MLB team would likely be successful in Utah based on the following:
1) Utah has seen strong population and economic growth, and teams in other sports like the Jazz have found success as a small market.
2) Utah would be geographically isolated from other MLB teams, with the closest being over 500 miles away.
3) The author's regression model predicts an MLB team in Utah would average around 16,600 fans per game, and over 22,000 fans per game over 10 years, which would be viable.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
This document provides information about graphs and graph theory concepts. It defines what a graph is consisting of vertices and edges. It describes different types of graphs such as undirected graphs, directed graphs, multigraphs, and pseudographs. It also discusses graph representations using adjacency matrices, adjacency lists, and incidence matrices. Additionally, it covers graph properties and concepts such as degrees of vertices, connected graphs, connected components, planar graphs, graph coloring, and the five color theorem.
This document provides an introduction and overview of graph theory. It defines some basic concepts such as vertices, edges, degrees of vertices, paths, walks, trees, and connectedness. It also introduces more advanced topics like isomorphic graphs, subgraphs, complements of graphs, bipartite graphs, and diameters of graphs. Finally, it discusses some applications of graph theory in fields like chemistry, physics, biology, computer science, operations research, maps, and the internet.
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.
This document discusses non split locating equitable domination in graphs. It begins with definitions of terms like domination number and non split locating equitable dominating set. It then presents several theorems that establish bounds on the non split locating equitable domination number of a graph based on its properties. These include bounds related to the number of vertices, minimum degree, number of pendant vertices, and whether the graph is regular or a tree. The document also characterizes the graphs that achieve equality in some of the bounds. In general, it analyzes the non split locating equitable domination number and relates it to other graph parameters.
This document defines basic concepts in graph theory. A graph consists of a set of vertices and edges connecting pairs of vertices. An adjacency matrix represents which vertices are connected by edges, while an incidence matrix represents which edges connect to each vertex. A simple graph cannot have loops or multiple edges between vertices. A complete graph connects each pair of vertices. Two graphs are isomorphic if there is a one-to-one correspondence between their vertices that preserves edge connections. Multigraphs allow multiple edges between vertices, while pseudographs also allow loops. The degree of a vertex is the number of edges connected to it.
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.
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 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.
1. Graph and Graph Terminologiesimp.pptxswapnilbs2728
There are five main categories of graphs: simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs. An undirected graph G consists of a set of vertices V and a set of edges E that connect the vertices. A directed graph consists of vertices V and directed edges E that have an initial and terminal vertex. There are several special types of simple graphs including complete graphs, cycles, wheels, and bipartite graphs.
This document defines and describes concepts related to fuzzy graphs and fuzzy digraphs. Key points include:
- A fuzzy graph is defined by two functions that assign membership values to vertices and edges.
- A fuzzy subgraph has lower or equal membership values for vertices and edges compared to the original graph.
- Effective edges and effective paths only include edges/paths where the membership equals the minimum vertex membership.
- Various graph measures are generalized to fuzzy graphs, such as vertex degree, order, size, and domination number.
- Fuzzy digraphs are defined similarly but with directed edges. Concepts like paths, independence, and domination are extended to fuzzy digraphs.
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 provides an overview of key concepts in graph theory, including:
- A graph consists of a set of vertices and edges connecting pairs of vertices.
- Paths and cycles are walks through a graph without repeating edges or vertices. A tree is an acyclic connected graph.
- The degree of a vertex is the number of edges connected to it. Regular graphs have all vertices of the same degree.
- Graphs can be represented using adjacency matrices and incidence matrices to show connections between vertices and edges.
- Directed graphs have edges oriented from a starting to ending vertex. Connectedness in directed graphs depends on the underlying graph or directionality of paths.
The document defines and describes various graph concepts and data structures used to represent graphs. It defines a graph as a collection of nodes and edges, and distinguishes between directed and undirected graphs. It then describes common graph terminology like adjacent/incident nodes, subgraphs, paths, cycles, connected/strongly connected components, trees, and degrees. Finally, it discusses two common ways to represent graphs - the adjacency matrix and adjacency list representations, noting their storage requirements and ability to add/remove nodes.
This document provides an introduction to graph theory, including basic terminology and concepts. It defines what a graph is mathematically as a collection of vertices and edges. It describes different types of graphs such as simple graphs, multiple graphs, weighted graphs, finite and infinite graphs, labeled graphs, and directed graphs. It also defines graph theory terms like adjacency, incidence, degree of a vertex, isomorphism, subgraphs, and graph operations like union and intersection. The document provides examples to illustrate each term and concept.
This document provides an introduction to graph theory, including basic terminology and concepts. It defines what a graph is mathematically as a collection of vertices and edges. It describes different types of graphs such as simple graphs, multiple graphs, weighted graphs, finite and infinite graphs, labeled graphs, and directed graphs. It also defines graph theory terms like adjacency, incidence, degree of a vertex, isomorphism, subgraphs, and graph operations like union and intersection. The document provides examples to illustrate each term and concept.
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.
Graph terminology and algorithm and tree.pptxasimshahzad8611
This document provides an overview of key concepts in graph theory including graph terminology, representations, traversals, spanning trees, minimum spanning trees, and shortest path algorithms. It defines graphs, directed vs undirected graphs, connectedness, degrees, adjacency, paths, cycles, trees, and graph representations using adjacency matrices and lists. It also describes breadth-first and depth-first traversals, spanning trees, minimum spanning trees, and algorithms for finding minimum spanning trees and shortest paths like Kruskal's, Prim's, Dijkstra's, Bellman-Ford and A* algorithms.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
1. G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian /
International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622
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Restrained Triple Connected Domination Number of a Graph
G. Mahadevan1 Selvam Avadayappan 2 V. G. Bhagavathi Ammal3
T. Subramanian4
1,4
Dept. of Mathematics, Anna University: Tirunelveli Region, Tirunelveli.
2
Dept.of Mathematics,VHNSN College, Virudhunagar.
3
PG Dept.of Mathematics, Sree Ayyappa College for Women, Chunkankadai, Nagercoil.
Abstract
The concept of triple connected graphs denotes its vertex set and E its edge set. Unless
with real life application was introduced in [9] by otherwise stated, the graph G has p vertices and q
considering the existence of a path containing edges. Degree of a vertex v is denoted by d(v), the
any three vertices of a graph G. In[3], G. maximum degree of a graph G is denoted by Δ(G).
Mahadevan et. al., was introduced the concept of We denote a cycle on p vertices by Cp, a path on p
triple connected domination number of a graph. vertices by Pp, and a complete graph on p vertices
In this paper, we introduce a new domination by Kp. A graph G is connected if any two vertices of
parameter, called restrained triple connected G are connected by a path. A maximal connected
domination number of a graph. A subset S of V subgraph of a graph G is called a component of G.
of a nontrivial graph G is called a dominating set The number of components of G is denoted by
of G if every vertex in V − S is adjacent to at least 𝜔(G). The complement 𝐺 of G is the graph with
one vertex in S. The domination number (G) of vertex set V in which two vertices are adjacent if
G is the minimum cardinality taken over all and only if they are not adjacent in G. A tree is a
dominating sets in G. A subset S of V of a connected acyclic graph. A bipartite graph (or
nontrivial graph G is called a restrained bigraph) is a graph whose vertex set can be divided
dominating set of G if every vertex in V − S is into two disjoint sets V1 and V2 such that every edge
adjacent to at least one vertex in S as well as has one end in V1 and another end in V2. A complete
another vertex in V - S. The restrained bipartite graph is a bipartite graph where every
domination number r(G) of G is the minimum vertex of V1 is adjacent to every vertex in V2. The
cardinality taken over all restrained dominating complete bipartite graph with partitions of order
sets in G. A subset S of V of a nontrivial graph G |V1|=m and |V2|=n, is denoted by Km,n. A star,
is said to be triple connected dominating set, if S denoted by K1,p-1 is a tree with one root vertex and p
is a dominating set and the induced sub graph – 1 pendant vertices. A bistar, denoted by B(m, n) is
<S> is triple connected. The minimum the graph obtained by joining the root vertices of the
cardinality taken over all triple connected stars K1,m and K1,n. The friendship graph, denoted
dominating sets is called the triple connected by Fn can be constructed by identifying n copies of
domination number and is denoted by 𝜸tc. A the cycle C3 at a common vertex. A wheel graph,
subset S of V of a nontrivial graph G is said to be denoted by Wp is a graph with p vertices, formed by
restrained triple connected dominating set, if S is a connecting a single vertex to all vertices of Cp-1. If S
restrained dominating set and the induced sub is a subset of V, then <S> denotes the vertex
graph <S> is triple connected. The minimum induced subgraph of G induced by S. The open
cardinality taken over all restrained triple neighbourhood of a set S of vertices of a graph G,
connected dominating sets is called the restrained denoted by N(S) is the set of all vertices adjacent to
triple connected domination number and is some vertex in S and N(S) ∪ S is called the closed
denoted by 𝜸rtc. We determine this number for neighbourhood of S, denoted by N[S]. A cut –
some standard graphs and obtain bounds for vertex (cut edge) of a graph G is a vertex (edge)
general graph. Its relationship with other graph whose removal increases the number of
theoretical parameters are also investigated. components. A vertex cut, or separating set of a
connected graph G is a set of vertices whose
Key words: Domination Number, Triple removal results in a disconnected. The connectivity
connected graph, Triple connected domination or vertex connectivity of a graph G, denoted by κ(G)
number, Restrained Triple connected domination (where G is not complete) is the size of a smallest
number. vertex cut. The chromatic number of a graph G,
AMS (2010): 05C69 denoted by χ(G) is the smallest number of colors
needed to colour all the vertices of a graph G in
1. Introduction which adjacent vertices receive different colours.
By a graph we mean a finite, simple, For any real number 𝑥, 𝑥 denotes the largest
connected and undirected graph G(V, E), where V integer less than or equal to 𝑥. A Nordhaus -
Gaddum-type result is a (tight) lower or upper
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2. G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian /
International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622
www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229
bound on the sum or product of a parameter of a Example 1.3 Let v1, v2, v3, v4, be the vertices of K4.
graph and its complement. Terms not defined here The graph K4(2P2, 2P2, 2P3, P3) is obtained from K4
are used in the sense of [2]. by attaching 2 times a pendant vertex of P2 on v1, 2
A subset S of V is called a dominating set times a pendant vertex of P2 on v2, 2 times a pendant
of G if every vertex in V − S is adjacent to at least vertex of P3 on v3 and 1 time a pendant vertex of P3
one vertex in S. The domination number (G) of G on v4 and is shown in Figure 1.1.
is the minimum cardinality taken over all
dominating sets in G. A dominating set S of a
connected graph G is said to be a connected
dominating set of G if the induced sub graph <S> is v1 v2
connected. The minimum cardinality taken over all
connected dominating sets is the connected v4 v3
domination number and is denoted by c.
A dominating set is said to be restrained
dominating set if every vertex in V – S is adjacent
to atleast one vertex in S as well as another vertex in
V - S. The minimum cardinality taken over all Figure 1.1 : K4(2P2, 2P2, 2P3, P3)
restrained dominating sets is called the restrained
domination number and is denoted by 𝛾r. 2. Restrained Triple connected domination
Many authors have introduced different number
types of domination parameters by imposing Definition 2.1 A subset S of V of a nontrivial graph
conditions on the dominating set [11, 12]. Recently, G is said to be a restrained triple connected
the concept of triple connected graphs has been dominating set, if S is a restrained dominating set
introduced by J. Paulraj Joseph et. al.,[9] by and the induced subgraph <S> is triple connected.
considering the existence of a path containing any The minimum cardinality taken over all restrained
three vertices of G. They have studied the properties triple connected dominating sets is called the
of triple connected graphs and established many restrained triple connected domination number of G
results on them. A graph G is said to be triple and is denoted by 𝛾rtc(G). Any triple connected two
connected if any three vertices lie on a path in G. dominating set with 𝛾rtc vertices is called a 𝛾rtc -
All paths, cycles, complete graphs and wheels are set of G.
some standard examples of triple connected graphs. Example 2.2 For the graphs G1, G2, G3 and G4, in
In[3], G. Mahadevan et. al., was introduced the Figure 2.1, the heavy dotted vertices forms the
concept of triple connected domination number of a restrained triple connected dominating sets.
graph. A subset S of V of a nontrivial graph G is said
to be a triple connected dominating set, if S is a
dominating set and the induced subgraph <S> is
triple connected. The minimum cardinality taken
over all triple connected dominating sets is called
the triple connected domination number of G and is
denoted by 𝛾tc(G). Any triple connected dominating
set with 𝛾tc vertices is called a 𝛾tc -set of G. In[4, 5,
6], G. Mahadevan et. al., was introduced
complementary triple connected domination
number, complementary perfect triple connected
domination number and paired triple connected
domination number of a graph and investigated
new results on them.
In this paper, we use this idea to develop
the concept of restrained triple connected
dominating set and restrained triple connected
domination number of a graph.
Theorem 1.1 [9] A tree T is triple connected if and
only if T ≅ Pp; p ≥ 3.
Notation 1.2 Let G be a connected graph with m
Figure 2.1 : Graph with 𝜸rtc = 3.
vertices v1, v2, …., vm. The graph obtained from G by
Observation 2.3 Restrained Triple connected
attaching n1 times a pendant vertex of 𝑃𝑙1 on the dominating set (rtcd set) does not exists for all
vertex v1, n2 times a pendant vertex of 𝑃𝑙2 on the graphs and if exists, then 𝛾rtc(G) ≥ 3.
vertex v2 and so on, is denoted by G(n1 𝑃𝑙1 , n2 𝑃𝑙2 ,
n3 𝑃𝑙3 , …., nm 𝑃𝑙 𝑚 ) where ni, li ≥ 0 and 1 ≤ i ≤ m.
226 | P a g e
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International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622
www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229
Example 2.4 For the graph G5, G6 in Figure 2.2, we Example 2.11 Consider the graph G8 and its
cannot find any restrained triple connected spanning subgraph H8 and G9 and its spanning
dominating set. subgraph H9 shown in Figure 2.4.
v3 v1 v2 v3
v1 v2
G8: H8:
v4 v6 v4 v5 v6
v5
v1 v2 v1 v2
Figure 2.2 : Graphs with no rtcd set
Throughout this paper we consider only connected
graphs for which triple connected two dominating
set exists. G9: H9:
v5 v5
Observation 2.5 The complement of the restrained
triple connected dominating set need not be a
restrained triple connected dominating set. v3 v4 v3 v4
Example 2.6 For the graph G7 in Figure 2.3, 𝑆 =
{v1, v4, v2} forms a restrained triple connected
dominating set of G3. But the complement V – S = Figure 2.4
{v3, v5} is not a restrained triple connected For the graph G8, S = {v1, v4, v2} is a restrained triple
dominating set. connected dominating set and so 𝛾rtc(G4) = 3. For
the spanning subgraph H8 of G8, S = {v1, v4, v2, v5} is
v2 a restrained triple connected dominating set so that
v1 𝛾rtc(H8) = 4. Hence 𝛾rtc(G8) < 𝛾rtc(H8). For the graph
G9, S = {v1, v2, v3} is a restrained triple connected
dominating set and so 𝛾rtc(G9) = 3. For the spanning
G7: v4 subgraph H9 of G9, S = {v1, v2, v3} is a restrained
v3 triple connected dominating set so that 𝛾rtc(H9) =
v5
3.Hence 𝛾rtc(G9) = 𝛾rtc(H9).
Theorem 2.12 For any connected graph G with p ≥
Figure 2.3 : Graph in which V–S is not a rtcd set 5, we have 3 ≤ 𝛾rtc(G) ≤ p - 2 and the bounds are
Observation 2.7 Every restrained triple connected sharp.
dominating set is a dominating set but not Proof The lower and bounds follows from
conversely. Definition 2.1. For K6, the lower bound is attained
Observation 2.8 Every restrained triple connected and for C9 the upper bound is attained.
dominating set is a connected dominating set but not Theorem 2.13 For any connected graph G with 5
conversely. vertices, 𝛾rtc(G) = p - 2 if and only if G ≅ K5, C5,
Exact value for some standard graphs: F2, K5 – e, K4(P2), C4(P2 ), C3(P3), C3(2P2 ) and any
1) For any cycle of order p ≥ 5, one of the following graphs given in Figure 2.5.
𝛾rtc(Cp) = 𝑝 − 2.
2) For any complete graph of order p
≥ 5, 𝛾rtc(Kp) = 3.
3) For any complete bipartite graph of
order p ≥ 5, 𝛾rtc(Km,n) = 3.
(where m, n ≥ 2 and m + n = p ).
Observation 2.9 If a spanning sub graph H of a
graph G has a restrained triple connected
dominating set, then G also has a restrained triple
connected dominating set.
Observation 2.10 Let G be a connected graph and
H be a spanning sub graph of G. If H has a
restrained triple connected dominating set, then
𝛾rtc(G) ≤ 𝛾rtc(H) and the bound is sharp.
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International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622
www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229
x x x
G1: y z G2: y z G3: y z
u v u v u v
x x x
G4: y z z
G5: y G6: y z
u v u v u v
x x x
G7: y z G8: y z z
G9: y
u v u v u v
Figure 2.5
Proof Suppose G is isomorphic to K5, C5, F2, K5 – e, Theorem 2.16 Let G be a graph such that G and 𝐺
K4(P2), C4(P2 ), C3(P3 ), C3(2P2) and any one of the have no isolates of order p ≥ 5. Then
given graphs in Figure 2.5., then clearly 𝛾rtc(G) = p (i) 𝛾rtc(G) + 𝛾rtc(𝐺 ) ≤ 2p -4
- 2. Conversely, Let G be a connected graph with 5 (ii) 𝛾rtc(G). 𝛾rtc(𝐺 ) ≤ (p – 2)2 and the bound is
vertices, and 𝛾rtc(G) = 3. Let S = {x, y, z} be the sharp.
𝛾rtc(G) –set of G. Take V – S = {u, v} and hence <V Proof The bound directly follows from Theorem
– S> = K2 = uv. 2.12. For cycle C5, both the bounds are attained.
Case (i) <S> = P3 = xyz.
Since G is connected, x (or equivalently z) 3 Relation with Other Graph Theoretical
is adjacent to u (or equivalently v) (or) y is adjacent Parameters
to u (or equivalently v). If x is adjacent to u. Since S Theorem 3.1 For any connected graph G with p ≥ 5
is a restrained triple connected dominating set, v is vertices, 𝛾rtc(G) + κ(G) ≤ 2p – 3 and the bound is
adjacent to x (or) y (or) z. If v is adjacent to z, then sharp if and only if G ≅ K5.
G ≅ C5. If v is adjacent to y, then G ≅ C4(P2). Now Proof Let G be a connected graph with p ≥ 5
by increasing the degrees of the vertices of K2 = uv, vertices. We know that κ(G) ≤ p – 1 and by
we have G ≅ G1 to G5, K5 – e, C3(P3). Now let y be Theorem 2.12, 𝛾rtc(G) ≤ p – 2. Hence 𝛾rtc(G) + κ(G)
adjacent to u. Since S is a restrained triple connected ≤ 2p – 3. Suppose G is isomorphic to K5. Then
dominating set, v is adjacent to x (or) y (or) z. If v is clearly 𝛾rtc(G) + κ(G) = 2p – 3. Conversely, Let
adjacent to y, then G ≅ C3(2P2). If v is adjacent to y 𝛾rtc(G) + κ(G) = 2p – 3. This is possible only if
and z, x is adjacent to z, then G ≅ K4(P2). Now by 𝛾rtc(G) = p – 2 and κ(G) = p – 1. But κ(G) = p – 1,
increasing the degrees of the vertices, we have G ≅ and so G ≅ Kp for which 𝛾rtc(G) = 3 = p – 2 . Hence
G6 to G8, C3(2P2). G ≅ K5.
Case (ii) <S> = C3 = xyzx. Theorem 3.2 For any connected graph G with p ≥ 5
Since G is connected, there exists a vertex in vertices, γrtc(G) + χ(G) ≤ 2p – 2 and the bound is
C3 say x is adjacent to u (or) v. Let x be adjacent to sharp if and only if G ≅ K5.
u. Since S is a restrained triple connected Proof Let G be a connected graph with p ≥ 5
dominating set, v is adjacent to x, then G ≅ F2. Now vertices. We know that 𝜒(G) ≤ p and by Theorem
by increasing the degrees of the vertices, we have G 2.12, 𝛾rtc(G) ≤ p – 2. Hence 𝛾rtc(G) + 𝜒(G) ≤ 2p – 2.
≅ G9, K5. In all the other cases, no new graph exists. Suppose G is isomorphic to K5. Then clearly 𝛾rtc(G)
The Nordhaus – Gaddum type result is given below: + 𝜒(G) = 2p – 2. Conversely, let 𝛾rtc(G) + 𝜒(G) =
228 | P a g e
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International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622
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2p – 2. This is possible only if 𝛾rtc(G) = p – 2 and Ciencia Indica, Vol. XXXI M, No. 2.: 847–
𝜒(G) = p. Since 𝜒(G) = p, G is isomorphic to Kp for 853.
which 𝛾rtc(G) = 3 = p – 2. Hence G ≅ K5. [11] Sampathkumar, E.; Walikar, HB (1979):
Theorem 3.3 For any connected graph G with p ≥ 5 The connected domination number of a
vertices, 𝛾rtc(G) + ∆(G) ≤ 2p – 3 and the bound is graph, J. Math. Phys. Sci 13 (6): 607–613.
sharp. [12] Teresa W. Haynes, Stephen T. Hedetniemi
Proof Let G be a connected graph with p ≥ 5 and Peter J. Slater (1998): Domination in
vertices. We know that ∆(G) ≤ p – 1 and by graphs, Advanced Topics, Marcel Dekker,
Theorem 2.12, 𝛾rtc(G) ≤ p. Hence 𝛾rtc(G) + ∆(G) ≤ New York.
2p – 3. For K5, the bound is sharp. [13] Teresa W. Haynes, Stephen T. Hedetniemi
and Peter J. Slater (1998): Fundamentals of
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