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.
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 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
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 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
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 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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The document 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.
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 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
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 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
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 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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The document discusses 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.
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.
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.
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.
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 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.
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 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 middle graph of a graph G, denoted by
M(G), is a graph whose vertex set is V(G)UE(G) , and two
vertices are adjacent if they are adjacent edges of G or one is a
vertex and other is a edge incident with it . The Line graph of G ,
written L(G), is the simple graph whose vertices are the edges of
G, with ef Є E(L(G)) when e and f have a common end vertex in
G. A set S of vertices of graph M(G) if S is an independent
dominating set of M(G) if S is an independent set and every
vertex not in S is adjacent to a vertex in S. The independent
middle domination number of G, denoted by iM(G) is the
minimum cardinality of an independent dominating set of
M(G).A dominating set D is a connected dominating set if <d> is
connected. The connected domination number, denoted by Ƴc , is
the minimum number of vertices in a connected dominating set.
In this paper many bounds on iL(G) ,iM(G),ƳM(G) were
obtained in terms of element of G, but not in terms of elements of
L(G) or M(G).
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.
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.
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.
IRJET - On Strong Blast Domination of GraphsIRJET Journal
This document introduces the concept of strong blast domination number of graphs. A strong blast dominating set S of a graph G is a connected dominating set such that the induced subgraph on S is triple connected, and each vertex in the graph but not in S is strongly dominated by a vertex in S. The document provides definitions, examples, and theorems about computing the strong blast domination number for various classes of graphs such as complete graphs, wheel graphs, and total graphs.
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.
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.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
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.
This document discusses different methodologies for access control and their interactions. It begins by introducing access control as a major security component for organizations to implement regulatory constraints. It then describes several common access control models in more detail, including Mandatory Access Control (MAC), Discretionary Access Control (DAC), and Role-Based Access Control (RBAC). MAC controls access based on a system-wide security policy, while DAC allows individual users some control over access permissions. The document analyzes advantages and limitations of each model and their suitability for different environments.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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.
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.
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.
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.
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 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.
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 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 middle graph of a graph G, denoted by
M(G), is a graph whose vertex set is V(G)UE(G) , and two
vertices are adjacent if they are adjacent edges of G or one is a
vertex and other is a edge incident with it . The Line graph of G ,
written L(G), is the simple graph whose vertices are the edges of
G, with ef Є E(L(G)) when e and f have a common end vertex in
G. A set S of vertices of graph M(G) if S is an independent
dominating set of M(G) if S is an independent set and every
vertex not in S is adjacent to a vertex in S. The independent
middle domination number of G, denoted by iM(G) is the
minimum cardinality of an independent dominating set of
M(G).A dominating set D is a connected dominating set if <d> is
connected. The connected domination number, denoted by Ƴc , is
the minimum number of vertices in a connected dominating set.
In this paper many bounds on iL(G) ,iM(G),ƳM(G) were
obtained in terms of element of G, but not in terms of elements of
L(G) or M(G).
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.
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.
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.
IRJET - On Strong Blast Domination of GraphsIRJET Journal
This document introduces the concept of strong blast domination number of graphs. A strong blast dominating set S of a graph G is a connected dominating set such that the induced subgraph on S is triple connected, and each vertex in the graph but not in S is strongly dominated by a vertex in S. The document provides definitions, examples, and theorems about computing the strong blast domination number for various classes of graphs such as complete graphs, wheel graphs, and total graphs.
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.
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.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
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.
This document discusses different methodologies for access control and their interactions. It begins by introducing access control as a major security component for organizations to implement regulatory constraints. It then describes several common access control models in more detail, including Mandatory Access Control (MAC), Discretionary Access Control (DAC), and Role-Based Access Control (RBAC). MAC controls access based on a system-wide security policy, while DAC allows individual users some control over access permissions. The document analyzes advantages and limitations of each model and their suitability for different environments.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document presents a novel fast-acting PI controller for regulating the dc-link voltage of a DSTATCOM (distribution static compensator). A DSTATCOM is used to mitigate power quality issues and compensate for nonlinear loads. Conventionally, a PI controller is used but has slow transient response. The paper proposes a fast-acting dc-link voltage controller based on the energy of the dc-link capacitor. It provides mathematical equations to design the gains of the conventional PI controller to achieve similar fast transient response as the proposed controller. Detailed simulations in MATLAB validate that the proposed controller has improved transient performance during load variations compared to the conventional controller.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document summarizes the use of data mining techniques in analyzing stock market data and foreign exchange rates. It discusses how data mining can be used to discover patterns and correlations in large financial datasets that may not be apparent to analysts. Specific techniques mentioned include neural networks, clustering, regression, and decision trees. Applications include predicting stock prices, detecting credit card fraud, and modeling foreign exchange markets. The document also notes some challenges in using data mining for financial data due to the dynamic nature of markets.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document discusses a study that examines the interrelationships between trust, perceived risk, and behavioral intention for technology acceptance and internet banking. The study develops an integrated model to explain how trust and perceived risk influence consumers' behavioral intention to use internet banking services. The research was conducted through a survey of 432 young Chinese consumers and analyzed the relationships between trust, perceived risk, and behavioral intention regarding the adoption of internet banking services in China.
This document summarizes and compares different methods for performing keyword searches in relational databases. It discusses candidate network-based methods, Steiner-tree based algorithms, and backward expanding keyword search approaches. It also evaluates methods that aim to improve search efficiency and accuracy, such as integrating multiple related tuple units and developing structure-aware indexes. The overall goal is to find an effective and efficient approach to keyword search over relational database structures.
This document summarizes a review of experimental and numerical investigations into friction stir welds of AA6063-T6 aluminum alloy. It begins with an abstract and introduction discussing friction stir welding as a solid-state joining process without melting. The amount of heat conducted into the workpiece determines weld quality. Understanding heat transfer is important to improve the process. Many studies have used simulation to determine temperature distribution under different welding conditions. The objective of this research was to develop a finite element simulation of AA6063-T6 aluminum alloy friction stir welding. Trend line equations would be developed to understand relationships between peak temperature and thermal conductivity, specific heat, and density. Tensile tests and hardness measurements were conducted on welded specimens
This document summarizes a review of experimental and numerical investigations into friction stir welds of AA6063-T6 aluminum alloy. It begins with an abstract and introduction discussing friction stir welding as a solid-state joining process without melting. The amount of heat conducted into the workpiece determines weld quality and tool life. The document then reviews various research using simulation to determine temperature distribution and develop relationships between input parameters and peak temperature. Experimental results for tensile strength and hardness of welded specimens are also reported and compared to simulation results.
El documento describe la segunda etapa de un ciclo de capacitación sobre ciencias en institutos de formación docente. Se detalla la organización del día, que incluye revisar productos del día anterior, trabajar en equipos en la redacción de un documento de gestión de clase y un instrumento de evaluación, y anticipar lo que sigue en el ciclo. Luego, se pide a los participantes redactar sugerencias para esos documentos y compartirlas con los demás, llevándose ejemplos para implementar.
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El documento proporciona información biográfica breve sobre varios escritores peruanos y latinoamericanos. Incluye detalles como su lugar y fecha de nacimiento, obras principales y en algunos casos detalles sobre su carrera literaria o fallecimiento.
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O documento apresenta os resultados operacionais e financeiros da Triunfo Participações e Investimentos no 2T11. O tráfego total de veículos aumentou 9,9% e a receita operacional bruta cresceu 25,9%. Entretanto, o lucro líquido caiu 95,4% devido ao aumento de 32,4% nas despesas financeiras. Os investimentos totais da empresa em 2011 somam R$2,2 bilhões, com foco no Terminal Portuário Brites.
Antonio Francisco Lisboa, conhecido como Aleijadinho, foi um importante escultor e arquiteto brasileiro do período colonial. Nasceu em 1730 em Ouro Preto e perdeu os dedos das mãos e pés aos 47 anos, tornando-se conhecido como "Aleijadinho". Suas principais obras incluem retábulos talhados em madeira e as estátuas da Via Sacra em Congonhas. Faleceu em 1814 em Ouro Preto.
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.
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.
1) Graph theory concepts such as undirected graphs, directed graphs, weighted graphs, mixed graphs, simple graphs, multigraphs, and pseudographs are introduced. Common graph terminology like vertices, edges, loops, degrees, adjacency and connectivity are defined.
2) Different types of graphs are discussed including trees, forests, cycles, wheels, complete graphs, bipartite graphs, and n-cubes. Matrix representations using adjacency matrices are also covered.
3) The document touches on graph isomorphism, Eulerian graphs, Hamiltonian graphs, and algorithms for determining graph properties like connectivity and number of components.
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 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.
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
This document discusses non split locating equitable domination in graphs. It begins with definitions of terms like domination number and non split locating equitable dominating set. It then presents several theorems that establish bounds on the non split locating equitable domination number of a graph based on its properties. These include bounds related to the number of vertices, minimum degree, number of pendant vertices, and whether the graph is regular or a tree. The document also characterizes the graphs that achieve equality in some of the bounds. In general, it analyzes the non split locating equitable domination number and relates it to other graph parameters.
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.
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.
The document provides definitions and examples related to graphs. It defines graphs, directed graphs, and terminology like vertices, edges, degrees, adjacency, and isomorphism. It discusses representations of graphs using adjacency lists and matrices. It also covers special types of graphs like trees, cycles, and bipartite graphs. Key concepts are illustrated with examples.
Application of Vertex Colorings with Some Interesting Graphsijtsrd
Firstly, basic concepts of graph and vertex colorings are introduced. Then, some interesting graphs with vertex colorings are presented. A vertex coloring of graph G is an assignment of colors to the vertices of G. And then by using proper vertex coloring, some interesting graphs are described. By using some applications of vertex colorings, two problems is presented interestingly. The vertex coloring is the starting point of graph coloring. The chromatic number for some interesting graphs and some results are studied. Ei Ei Moe "Application of Vertex Colorings with Some Interesting Graphs" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29263.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/29263/application-of-vertex-colorings-with-some-interesting-graphs/ei-ei-moe
The document discusses various graph theory concepts including:
- Types of graphs such as simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs which differ based on allowed edge connections.
- Graph terminology including vertices, edges, degrees, adjacency, incidence, paths, cycles, and representations using adjacency lists and matrices.
- Weighted graphs and algorithms for finding shortest paths such as Dijkstra's algorithm.
- Euler and Hamilton paths/circuits and conditions for their existence.
- The traveling salesman problem of finding the shortest circuit visiting all vertices.
The document describes results from graph theory, specifically matching theory, flows, coloring, and combinatorial nullstellensatz. It investigates classical theorems like Hall's theorem, Tutte's theorem, Dilworth's theorem, and Nash-Williams theorem on matchings in bipartite and general graphs. It also discusses network flows and Baranyai's theorem. The problems were taken from exercises in a book on modern graph theory.
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.
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.
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.
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.
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
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.
1. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
Complementary Perfect Triple Connected Domination Number of
a Graph
G. Mahadevan*, Selvam Avadayappan**, A.Mydeen bibi***,
T.Subramanian****
*
Dept. of Mathematics, Anna University of Technology, Tirunelveli-627 002.
**
Dept.of Mathematics, VHNSN College, Virudhunagar.
***
Research Scholar, Mother Teresa Women’s University, Kodaikanal.
****
Research Scholar, Anna University of Technology Tirunelveli, Tirunelveli.
Abstract: The number of components of G is denoted by 𝜔 (G).
The concept of triple connected graphs The complement 𝐺 of G is the graph with vertex set
with real life application was introduced in [9] by V in which two vertices are adjacent if and only if
considering the existence of a path containing any they are not adjacent in G. A tree is a connected
three vertices of G. A graph G is said to be triple acyclic graph. A bipartite graph (or bigraph) is a
connected if any three vertices lie on a path in G. graph whose vertices can be divided into two disjoint
In [3], the concept of triple connected dominating sets U and V such that every edge connects a vertex
set was introduced. A set S V is a triple in U to one in V. A complete bipartite graph is a
connected dominating set if S is a dominating set special kind of bipartite graph where every vertex of
of G and the induced sub graph <S> is triple the first set is connected to every vertex of the second
connected. The triple connected domination set. The complete bipartite graph with partitions of
number 𝛾 tc(G) is the minimum cardinality taken order |V1|=m and |V2|=n, is denoted Km,n. A star,
over all triple connected dominating sets in G. In denoted by K1,p-1 is a tree with one root vertex and p –
this paper, we introduce a new domination 1 pendant vertices. A bistar, denoted by B(m, n) is
parameter, called Complementary perfect triple the graph obtained by joining the root vertices of the
connected domination number of a graph. A set stars K1,m and K1,n.
S V is a complementary perfect triple connected The friendship graph, denoted by Fn can be
dominating set if S is a triple connected constructed by identifying n copies of the cycle C3 at
dominating set of G and the induced sub graph a common vertex. A wheel graph, denoted by Wp is a
<V - S> has a perfect matching. The graph with p vertices, formed by connecting a single
complementary perfect triple connected vertex to all vertices of an (p-1) cycle. A helm graph,
denoted by Hn is a graph obtained from the wheel Wn
domination number 𝛾 cptc(G) is the minimum by joining a pendant vertex to each vertex in the
cardinality taken over all complementary perfect outer cycle of Wn by means of an edge. Corona of
triple connected dominating sets in G. We two graphs G1 and G2, denoted by G1 G2 is
determine this number for some standard classes
of graphs and obtain some bounds for general the disjoint union of one copy of G1 and |V1| copies of
graph. Their relationships with other graph G2 (|V1| is the number of vertices in G1) in which ith
theoretical parameters are investigated. vertex of G1 is joined to every vertex in the ith copy
of G2. For any real number , denotes the largest
Key words: Domination Number, Triple connected integer less than or equal to . If S is a subset of V,
graph, Complementary perfect triple connected then <S> denotes the vertex induced subgraph of G
domination number. induced by S. The open neighbourhood of a set S of
AMS (2010): 05C 69 vertices of a graph G, denoted by N(S) is the set of all
vertices adjacent to some vertex in S and N(S) S is
1 Introduction
By a graph we mean a finite, simple, called the closed neighbourhood of S, denoted by
connected and undirected graph G (V, E), where V N[S]. The diameter of a connected graph is the
denotes its vertex set and E its edge set. Unless maximum distance between two vertices in G and is
otherwise stated, the graph G has p vertices and q denoted by diam (G). A cut – vertex (cut edge) of a
edges. Degree of a vertex v is denoted by d (v), the graph G is a vertex (edge) whose removal increases
maximum degree of a graph G is denoted by Δ(G). the number of components. A vertex cut, or
We denote a cycle on p vertices by Cp, a path on p separating set of a connected graph G is a set of
vertices by Pp, and a complete graph on p vertices by vertices whose removal renders G disconnected. The
Kp. A graph G is connected if any two vertices of G connectivity or vertex connectivity of a graph G,
are connected by a path. A maximal connected denoted by κ(G) (where G is not complete) is the size
subgraph of a graph G is called a component of G. of a smallest vertex cut. A connected subgraph H of a
260 | P a g e
2. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
connected graph G is called a H -cut if (G – H) ≥ Theorem 1.1 [9] A tree T is triple connected if and
2. The chromatic number of a graph G, denoted by only if T Pp; p ≥ 3.
χ(G) is the smallest number of colors needed to Theorem 1.2 [9] A connected graph G is not triple
colour all the vertices of a graph G in which adjacent connected if and only if there exists a H -cut with
vertices receive different colour. Terms not defined (G – H) ≥ 3 such that = 1 for at
here are used in the sense of [2].
least three components C1, C2, and C3 of G – H.
A subset S of V is called a dominating set of
Notation 1.3 Let G be a connected graph with m
G if every vertex in V − S is adjacent to at least one
vertices v1, v2, …., vm. The graph G(n1 , n2 ,
vertex in S. The domination number (G) of G is the
minimum cardinality taken over all dominating sets n3 , …., nm ) where ni, li ≥ 0 and 1 ≤ i ≤ m, is
in G. A dominating set S of a connected graph G is
obtained from G by attaching n1 times a pendant
said to be a connected dominating set of G if the
induced sub graph <S> is connected. The minimum vertex of on the vertex v1, n2 times a pendant
cardinality taken over all connected dominating set is vertex of on the vertex v2 and so on.
the connected domination number and is denoted by
Example 1.4 Let v1, v2, v3, v4, be the vertices of K4,
c. A subset S of V of a nontrivial graph G is said to
the graph K4 (2P2, P3, P4, P3) is obtained from K4 by
be Complementary perfect dominating set, if S is a
dominating set and the subgraph induced by <V-S> attaching 2 times a pendant vertex of P 2 on v1, 1 time
has a perfect matching. The minimum cardinality a pendant vertex of P3 on v2, 1 times a pendant vertex
of P4 on v3 and 1 time a pendant vertex of P 3 on v4
taken over all Complementary perfect dominating
and is shown in Figure 1.1.
sets is called the Complementary perfect domination
number and is denoted by cp.
One can get a comprehensive survey of
results on various types of domination number of a v1 v2
graph in [11].
Many authors have introduced different
types of domination parameters by imposing v4 v3
conditions on the dominating set [1,8]. Recently the
concept of triple connected graphs was introduced by
Paulraj Joseph J. et. al.,[9] by considering the
Figure 1.1: K4 (2P2, P3, P4, P3)
existence of a path containing any three vertices of G.
They have studied the properties of triple connected
graph and established many results on them. A graph 2 Complementary Perfect Triple Connected
G is said to be triple connected if any three vertices Domination Number
lie on a path in G. All paths and cycles, complete Definition 2.1 A set S V is a complementary
graphs and wheels are some standard examples of perfect triple connected dominating set if S is a triple
triple connected graphs. connected dominating set of G and the induced sub
A set S V is a triple connected dominating graph <V - S> has a perfect matching. The
set if S is a dominating set of G and the induced sub complementary perfect triple connected domination
graph <S> is triple connected. The triple connected number cptc(G) is the minimum cardinality taken
domination number tc(G) is the minimum over all complementary perfect triple connected
cardinality taken over all triple connected dominating dominating sets in G. Any triple connected
sets in G. dominating set with cptc vertices is called a cptc -set
A set S V is a complementary perfect of G.
triple connected dominating set if S is a triple Example 2.2 For the graph G1 in Figure 2.1, S = {v1,
connected dominating set of G and the induced sub v2, v5} forms a cptc -set of G1. Hence cptc (G1) = 3.
graph <V - S> has a perfect matching. The
complementary perfect triple connected domination
v1
number cptc(G) is the minimum cardinality taken
over all complementary perfect triple connected
dominating sets in G.
G 1: v5 v2
In this paper we use this idea to develop the
concept of complementary perfect triple connected v3
v4
dominating set and complementary perfect triple
connected domination number of a graph. Figure 2.1
Observation 2.3 Complementary perfect triple
connected dominating set does not exists for all
Previous Results graphs and if exists, then cptc ≥ 3.
261 | P a g e
3. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
Example 2.4 For the graph G2 in Figure 2.2, any
v6 v1
minimum dominating set must contain the supports
and any connected dominating set containing these
supports is not Complementary perfect triple
connected and hence cptc does not exists. K6: v5 v2
G 2:
v4 v3
Figure 2.2 Figure 2.5
Remark 2.5 Throughout this paper we consider only Theorem 2.12 If the induced subgraph of each
connected graphs for which complementary perfect connected dominating set of G has more than two
triple connected dominating set exists. pendant vertices, then G does not contain a
Observation 2.6 The complement of the Complementary perfect triple connected dominating
complementary perfect triple connected dominating set.
set need not be a complementary perfect triple Proof This theorem follows from Theorem 1.2.
connected dominating set. Example 2.13 For the graph G6 in Figure 2.6,
Example 2.7 For the graph G3 in Figure 2.3, S = {v6, v2, v3, v4} is a minimum connected
= {v1, v2, v3} forms a Complementary perfect triple dominating set so that c (G6) = 4. Here we notice
connected dominating set of G3. But the complement that the induced subgraph of S has three pendant
V – S = {v4, v5, v6, v7} is not a Complementary perfect vertices and hence G does not have a Complementary
triple connected dominating set. perfect triple connected dominating set.
v7
v1 v3
v2 v6
G 6:
v7
v4 v1 v2
v6 v3 v4 v5
G 3: v5
Figure 2.6
Figure 2.3
Observation 2.8 Every Complementary perfect triple Complementary perfect triple connected
connected dominating set is a dominating set but not domination number for some standard graphs are
the converse. given below.
Example 2.9 For the graph G4 in Figure 2.4, S = {v1} 1) For any cycle of order p ≥ 5, cptc (Cp) = p – 2.
is a dominating set, but not a complementary perfect 2) For any complete bipartite graph of order p ≥ 5,
triple connected dominating set.
cptc (Km,n) =
v1 (where m, n ≥ 2 and m + n = p.
3) For any complete graph of order p ≥ 5,
G 4: v4
v2 cptc (Kp) =
v3 4) For any wheel of order p ≥ 5,
Figure 2.4
Observation 2.10 For any connected graph G, (G) cptc (Wp) =
≤ cp(G) ≤ cptc(G) and the inequalities are strict. 5) For any Fan graph of order p ≥ 5,
Example 2.11 For K6 in Figure 2.5, (K6) = {v1} = 1,
cp(K6) = {v1, v2}= 2 and cptc(K6) = {v1, v2, v3, v4}= cptc (Fp) =
4. Hence (G) ≤ cp(G) ≤ cptc(G). 6) For any Book graphs of order p ≥ 6,
cptc (Bp) = 4.
7) For any Friendship graphs of order p ≥ 5,
cptc (Fn)) = 3.
Observation 2.14 If a spanning sub graph H of a
graph G has a Complementary perfect triple
262 | P a g e
4. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
connected dominating set, then G also has a Proof The lower and upper bounds trivially follows
complementary perfect triple connected dominating from Definition 2.1. For C5, the lower bound is
set. attained and for C9 the upper bound is attained.
Example 2.16 For any graph G and H in Figure 2.7, Theorem 2.20 For a connected graph G with 5
S = {v1, v4, v5} is a complementary perfect triple vertices, cptc(G) = p – 2 if and only if G is
connected dominating set and so cptc (G) = 3. For the isomorphic to C5, W5, K5, K2,3, F2, K5 – {e}, K4(P2),
spanning subgraph H, S = {v1, v4, v5} is a C4(P2), C3(P3), C3(2P2) or any one of the graphs
complementary perfect triple connected dominating shown in Figure 2.9.
set and so cptc(H) = 3. H1: x H2: x
V1
G:
v y v y
V5
V4 V2 z u z u
H3: y H4: y
V3
H: V1 x z x u
u v z v
V4
V5 V2 H5: x H6: x
v y v y
V3 u z u z
Figure 2.7
H7: x
Observation 2.17 Let G be a connected graph and H
be a spanning sub graph of G. If H has a
complementary perfect triple connected dominating
v y
set, then cptc (G) ≤ cptc (H) and the bound is sharp.
Example 2.18 For the graph G in Figure 2.8, u z
S = {v1, v2, v7, v8} is a complementary perfect triple
connected dominating set and so cptc (G) = 4. For the Figure 2.9
spanning subgraph H of G, S = {v1, v2, v3, v6, v7, v8} Proof Suppose G is isomorphic to C5, W5, K5, K2,3,
is a complementary perfect triple connected F2, K5 – {e}, K4(P2), C4(P2), C3(P3), C3(2P2) or any
dominating set and so cptc (H) = 6. one of the graphs H1 to H7 given in Figure 2.9, then
clearly cptc (G) = p – 2. Conversely, let G be a
G: V1 V2 V3 V4
connected graph with 5 vertices and cptc (G) = 3.
Let S = {x, y, z} be a cptc -set, then clearly <S> = P3
or C3. Let V – S = V (G) – V(S) = {u, v}, then
<V – S> = K2 = uv.
Case (i) <S> = P3 = xyz.
Since G is connected, there exists a vertex say x (or y,
V8 V7 V6 V5 z) in P3 is adjacent to u (or v) in K2, then cptc -set of G
does not exists. But on increasing the degrees of the
7
V1 V2 V3 V4 vertices of S, let x be adjacent to u and z be adjacent
H:
to v. If d(x) = d(y) = d(z) = 2, then G C5. Now by
increasing the degrees of the vertices, by the above
argument, we have G K5, K5 – {e}, K4(P2), C4(P2),
C3(P3) or any one of the graphs H1 to H7 given in
Figure 2.9. Since G is connected, there exists a vertex
say y in P3 is adjacent to u (or v) in K2, then cptc -set
V8 V7 V6 V5
of G does not exists. But on increasing the degrees of
Figure 2.8 the vertices of S, let y be adjacent to v, x be adjacent
Theorem 2.19 For any connected graph G with p ≥ 5, to u and v and z be adjacent to u and v. If d(x) = 3,
we have 3 cptc(G) ≤ p - 2 and the bounds are sharp.
263 | P a g e
5. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
d(y) = 3, d(z) = 3, then G W5. Now by increasing 3. Complementary Perfect Triple Connected
the degrees of the vertices, by the above argument, Domination Number and Other Graph
we have G K2,3, C3(2P2). In all the other cases, no Theoretical Parameters
new graph exists. Theorem 3.1 For any connected graph G with p ≥ 5
Case (ii) <S> = C3 = xyzx. vertices, cptc (G) + κ(G) ≤ 2p – 3 and the bound is
Since G is connected, there exists a vertex say x (or y, sharp if and only if G K5.
z) in C3 is adjacent to u (or v) in K2, then S = {x, u, v} Proof Let G be a connected graph with p ≥ 5 vertices.
forms a cptc -set of G so that cptc (G) = p – 2. If We know that κ(G) ≤ p – 1 and by Theorem 2.19, cptc
d(x) = 3, d(y) = d(z) = 2, then G C3(P3). If d(x) = 4, (G) ≤ p – 2. Hence cptc (G) + κ(G) ≤ 2p – 3. Suppose
d(y) = d(z) = 2, then G F2. In all the other cases, no G is isomorphic to K5. Then clearly cptc (G) + κ (G)
new graph exists. = 2p – 3. Conversely, Let cptc (G) + κ(G) = 2p – 3.
Nordhaus – Gaddum Type result: This is possible only if cptc (G) = p – 2 and κ (G) =
Theorem 2.21 Let G be a graph such that G and p – 1. But κ (G) = p – 1, and so G Kp for which cptc
have no isolates of order p ≥ 5, then (G) = 3 = p – 2 so that p = 5. Hence G K5.
(i)cptc (G) + cptc ( ) ≤ 2(p – 2) Theorem 3.2 For any connected graph G with p ≥ 5
(ii) cptc (G) . cptc ( )≤ (p – 2)2 and the bounds are vertices, cptc (G) + (G) ≤ 2p – 2 and the bound is
sharp if and only if G K5.
sharp. Proof Let G be a connected graph with p ≥ 5 vertices.
Proof The bounds directly follow from Theorem
We know that (G) ≤ p and by Theorem 2.19,
2.19. For the cycle C5, cptc (G) + cptc ( ) = 2(p – 2) cptc(G) ≤ p – 2. Hence cptc (G) + (G) ≤ 2p – 2.
and cptc (G). cptc ( )≤ (p – 2)2. Suppose G is isomorphic to K5. Then clearly
Theorem 2.22 γcptc (G) ≥ p / ∆+1 cptc (G) + (G) = 2p – 2. Conversely, Let cptc (G)
Proof Every vertex in V-S contributes one to degree + (G) = 2p – 2. This is possible only if cptc (G) =
sum of vertices of S. Then |V - S| u S d(u) where p – 2 and (G) = p. But (G) = p, and so G is
S is a complementary perfect triple connected isomorphic to Kp for which cptc (G) = 3 = p – 2 so
dominating set. So |V - S| γcptc ∆ which implies that p = 5. Hence G K5.
(|V|- |S|) γcptc ∆. Therefore p - γcptc γcptc ∆, which Theorem 3.3 For any connected graph G with p ≥ 5
implies γcptc (∆+1) ≥ p. Hence γcptc (G) ≥ p / ∆+1 vertices, cptc (G) +(G) ≤ 2p – 3 and the bound is
Theorem 2.23 Any complementary perfect triple sharp if and only if G is isomorphic to W5, K5, C3(2),
connected dominating set of G must contains all the K5 – {e}, K4(P2), C3(2P2) or any one of the graphs
pendant vertices of G. shown in Figure 3.1.
Proof Let S be any complementary perfect triple
connected dominating set of G. Let v be a pendant G 1: v4 G 2: v4
vertex with support say u. If v does not belong to S,
then u must be in S, which is a contradiction S is a
complementary perfect triple connected dominating v3 v v3 v
set of G. Since v is a pendant vertex, so v belongs to
S. v2 v1 v2 v1
Observation 2.24 There exists a graph for which G 3: v4 v4
γcp (G) = γtc(G) = γcptc (G) is given below.
G 4:
v1 v2 v3 v v3 v
v6
v2 v1 v2 v1
v5 Figure 3.1
Proof Let G be a connected graph with p ≥ 5 vertices.
v4
We know that (G) ≤ p – 1 and by Theorem 2.19,
cptc (G) ≤ p – 2. Hence cptc (G) + (G) ≤ 2p – 3. Let
v3 v7 G be isomorphic to W5, K5,C3(2), K5 – {e}, K4(P2),
C3(2P2) or any one of the graphs G1 to G4 given in
H: Figure 3.1, then clearly cptc (G) +(G) = 2p – 3.
Figure 2.10 Conversely, Let cptc (G) + (G) = 2p – 3. This is
For the graph Hi in figure 2.10, S = {v1, v3, v7} is a possible only if cptc (G) = p – 2 and (G) = p – 1.
triple connected dominating set complementary But cptc (G) = p – 2 and (G) = p – 1, by
perfect and complementary perfect triple connected Theorem 2.20, we have G W5, K5, C3(2), K5 – {e},
dominatin set. Hence γcp (G) = γtc(G) = γcptc (G) = 3. K4(P2), C3(2P2) and the graphs G1 to G4 given in
Figure 3.1.
264 | P a g e
6. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
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