Lection is one of the course "Discrete Mathematics." More detailed information about the programs of the Economic Cybernetics Department SumDU you could find here: https://ek.biem.sumdu.edu.ua/courses
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.
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.
This document provides an overview of graph theory concepts. It defines what a graph is consisting of vertices and edges. It describes different types of graphs such as directed vs undirected, simple vs complex graphs. It introduces common graph terminology like degree of a vertex, adjacent/incident vertices, and connectivity. Examples of applications are given such as transportation networks, web graphs, and scheduling problems. Special graph cases like complete graphs and cycles are also defined.
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. A graph is a collection of objects called vertices that are connected by links called edges. Graphs can be represented as a pair of sets (V,E) where V is the set of vertices and E is the set of edges.
2. There are several important terms used to describe graphs including adjacent nodes, degree of a node, regular graphs, paths, cycles, connected graphs, and complete graphs. Graphs can be represented using adjacency matrices or adjacency lists.
3. There are two main techniques for traversing graphs - depth-first search (DFS) and breadth-first search (BFS). DFS uses a stack and traverses graphs in a depth-wise manner while BFS uses a
This document provides an overview of graph theory. It defines various graph types including simple graphs, multigraphs, pseudographs, directed graphs, and labeled graphs. It also defines key graph terminology such as vertices, edges, degree, adjacency, connectivity, and planar graphs. Graph theory has many applications in fields like transportation, computer networks, and chemistry for modeling relationships between objects.
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
This document defines and provides examples of different types of graphs, including finite and infinite graphs, simple graphs, complete graphs, bipartite graphs, and regular graphs. It introduces key graph terminology like vertices, edges, degrees, adjacency, and isolation. Examples are provided to illustrate concepts like the handshake theorem, determining if certain degree sequences can form graphs, and drawing regular 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.
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.
This document provides an overview of graph theory concepts. It defines what a graph is consisting of vertices and edges. It describes different types of graphs such as directed vs undirected, simple vs complex graphs. It introduces common graph terminology like degree of a vertex, adjacent/incident vertices, and connectivity. Examples of applications are given such as transportation networks, web graphs, and scheduling problems. Special graph cases like complete graphs and cycles are also defined.
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. A graph is a collection of objects called vertices that are connected by links called edges. Graphs can be represented as a pair of sets (V,E) where V is the set of vertices and E is the set of edges.
2. There are several important terms used to describe graphs including adjacent nodes, degree of a node, regular graphs, paths, cycles, connected graphs, and complete graphs. Graphs can be represented using adjacency matrices or adjacency lists.
3. There are two main techniques for traversing graphs - depth-first search (DFS) and breadth-first search (BFS). DFS uses a stack and traverses graphs in a depth-wise manner while BFS uses a
This document provides an overview of graph theory. It defines various graph types including simple graphs, multigraphs, pseudographs, directed graphs, and labeled graphs. It also defines key graph terminology such as vertices, edges, degree, adjacency, connectivity, and planar graphs. Graph theory has many applications in fields like transportation, computer networks, and chemistry for modeling relationships between objects.
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
This document defines and provides examples of different types of graphs, including finite and infinite graphs, simple graphs, complete graphs, bipartite graphs, and regular graphs. It introduces key graph terminology like vertices, edges, degrees, adjacency, and isolation. Examples are provided to illustrate concepts like the handshake theorem, determining if certain degree sequences can form graphs, and drawing regular graphs.
This document defines and explains basic graph terminology and representations. It defines simple and directed graphs, and discusses graph models using examples of railway and sports tournament networks. Key graph concepts covered include adjacency, edges, degrees, isolated and pendant vertices, and handshaking theorem. It also explains how to represent graphs using adjacency matrices and incidence matrices.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices and edges connecting pairs of vertices. There are many types of graphs including trees, which are connected acyclic graphs. Spanning trees are subgraphs of a graph that connect all vertices using the minimum number of edges. Key concepts in graph theory include paths, connectedness, cycles, and isomorphism between graphs.
Graphs can be used to visually represent numerical data or relationships between objects. There are different types of graphs like bar graphs, pictographs, and line graphs that display data in various ways. Graphs have advantages like showing qualitative behavior and overall features of data at a glance, but may lack accuracy. The document then discusses different types of graphs like directed/undirected graphs and symmetric/asymmetric graphs in more detail.
Graphs can be used to visually represent numerical data or relationships between objects. There are different types of graphs like bar graphs, pictographs, and line graphs that display data in various ways. Graphs have advantages like showing qualitative behavior and overall features of data at a glance, but may lack accuracy. The document then discusses different types of graphs like directed/undirected graphs and symmetric/asymmetric graphs in more detail.
Graphs are used to visually represent numerical data or relationships between objects. There are different types of graphs including bar graphs, pictographs, and digraphs. Bar graphs display data using rectangular bars to make comparisons easy to see. Pictographs use pictures to represent quantities. Digraphs are directed graphs represented by nodes connected by arrows to model relationships. Some key aspects are then described in more detail such as symmetric and asymmetric graphs, and examples are provided.
Graph in data structure it gives you the information of the graph application. How to represent the Graph and also Graph Travesal is also there many terms are there related to garph
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relationships between objects. A graph consists of vertices and edges that connect the vertices. Graph theory is used in computer science to model problems that can be represented as networks, such as determining routes in a city or designing circuits. It allows analyzing properties of graphs like connectivity and determining the minimum number of resources required. A graph can be represented using an adjacency matrix or adjacency list to store information about its vertices and edges.
A perfect matching in a bipartite graph is a matching that matches all vertices. The document discusses algorithms for finding a perfect matching in regular bipartite graphs. The currently most efficient algorithm takes time O(m), where m is the number of edges and n is the number of vertices. The document improves this to O(min{m, n2.5ln n/d}) by proving a uniform sampling theorem: sampling each edge independently with probability O(n ln n/d2) results in a graph that has a perfect matching with high probability.
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.
The document discusses graphs and graph algorithms. It begins by defining what a graph is - a collection of vertices connected by edges. It then lists four learning objectives related to representing graphs, traversing graphs, calculating minimum spanning trees, and finding shortest routes. The document goes on to describe different ways of representing graphs through adjacency matrices and lists. It also explains graph traversal algorithms like depth-first search and breadth-first search. Finally, it discusses algorithms for finding minimum spanning trees and shortest paths in weighted graphs.
This document discusses various graph parameters related to distance between vertices in a graph, including diameter, radius, and average distance. It examines how these parameters are defined, how they relate to each other and other graph properties, and how they behave in different graph classes. It also discusses characterizations of graph classes based on distance properties and generalizations to concepts like Steiner trees.
The document discusses graphs and graph algorithms. It defines what a graph is and how they can be represented. It also explains graph traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). Additionally, it covers algorithms for finding the shortest path using Dijkstra's algorithm and calculating minimum spanning trees using Kruskal's algorithm.
This document provides information about graphs and graph algorithms. It discusses different graph representations including adjacency matrices and adjacency lists. It also describes common graph traversal algorithms like depth-first search and breadth-first search. Finally, it covers minimum spanning trees and algorithms to find them, specifically mentioning Kruskal's algorithm.
This document is a project report submitted by S. Manikanta in partial fulfillment of the requirements for a Master of Science degree in Mathematics. The report discusses applications of graph theory. It provides an overview of graph theory concepts such as definitions of graphs, terminology used in graph theory, different types of graphs, trees and forests, graph isomorphism and operations, walks and paths in graphs, representations of graphs using matrices, applications of graphs in areas like computer science, fingerprint recognition, security, and more. The document also includes examples and illustrations to explain various graph theory concepts.
Unit II_Graph.pptxkgjrekjgiojtoiejhgnltegjtepournima055
The document discusses graphs and graph algorithms. It covers basic graph concepts like vertices, edges, paths, cycles. It describes different ways to represent graphs like adjacency matrix, adjacency list and their pros and cons. It also discusses operations on graphs like inserting and deleting vertices and edges. The document explains traversal algorithms like depth-first search and breadth-first search. It covers minimum spanning tree algorithms like Prim's and Kruskal's. It also briefly discusses shortest path algorithms like Dijkstra's and topological sorting.
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.
This document provides an introduction to topological graph theory. It begins with definitions of basic graph theory concepts from a topological perspective, such as representing graphs with curved arcs instead of straight lines. It then discusses graph drawings, incidence matrices, vertex valence, and graph maps/isomorphisms. Important classes of graphs are introduced, such as trees, paths, cycles, and complete graphs. The document aims to introduce preliminary concepts of topological graph theory to students in a simple manner.
The document discusses Hamiltonian graphs, which are graphs that contain a Hamiltonian circuit or path. A Hamiltonian circuit visits each vertex in the graph exactly once and forms a cycle. The document provides an example of a graph that is Hamiltonian but not Eulerian, and vice versa. It notes that while it is clear only connected graphs can be Hamiltonian, there is no simple criterion to determine if a graph is Hamiltonian like there is for Eulerian graphs. It presents Dirac's sufficient condition for a graph to be Hamiltonian.
This document defines and describes different types of graphs. It begins by defining what a graph is as consisting of vertices and edges. It then discusses finite vs infinite graphs, isolated and pendant vertices, null graphs, regular graphs, subgraphs, complete graphs, bipartite graphs, and complete bipartite graphs. Diagrams are provided to illustrate each graph type.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
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journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
Lection is one of the course "Discrete Mathematics." More detailed information about the programs of the Economic Cybernetics Department SumDU you could find here: https://ek.biem.sumdu.edu.ua/courses
This document discusses tree data structures and binary search trees. It begins with definitions of trees and their components like nodes, edges, and root/child relationships. It then covers tree traversals like preorder, inorder and postorder. Next, it defines binary search trees and their properties for storing and retrieving data efficiently. It provides examples of inserting, searching for, and deleting nodes from binary search trees. Searching and insertion are straightforward, while deletion has three cases depending on the number of child nodes.
This document defines and explains basic graph terminology and representations. It defines simple and directed graphs, and discusses graph models using examples of railway and sports tournament networks. Key graph concepts covered include adjacency, edges, degrees, isolated and pendant vertices, and handshaking theorem. It also explains how to represent graphs using adjacency matrices and incidence matrices.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices and edges connecting pairs of vertices. There are many types of graphs including trees, which are connected acyclic graphs. Spanning trees are subgraphs of a graph that connect all vertices using the minimum number of edges. Key concepts in graph theory include paths, connectedness, cycles, and isomorphism between graphs.
Graphs can be used to visually represent numerical data or relationships between objects. There are different types of graphs like bar graphs, pictographs, and line graphs that display data in various ways. Graphs have advantages like showing qualitative behavior and overall features of data at a glance, but may lack accuracy. The document then discusses different types of graphs like directed/undirected graphs and symmetric/asymmetric graphs in more detail.
Graphs can be used to visually represent numerical data or relationships between objects. There are different types of graphs like bar graphs, pictographs, and line graphs that display data in various ways. Graphs have advantages like showing qualitative behavior and overall features of data at a glance, but may lack accuracy. The document then discusses different types of graphs like directed/undirected graphs and symmetric/asymmetric graphs in more detail.
Graphs are used to visually represent numerical data or relationships between objects. There are different types of graphs including bar graphs, pictographs, and digraphs. Bar graphs display data using rectangular bars to make comparisons easy to see. Pictographs use pictures to represent quantities. Digraphs are directed graphs represented by nodes connected by arrows to model relationships. Some key aspects are then described in more detail such as symmetric and asymmetric graphs, and examples are provided.
Graph in data structure it gives you the information of the graph application. How to represent the Graph and also Graph Travesal is also there many terms are there related to garph
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relationships between objects. A graph consists of vertices and edges that connect the vertices. Graph theory is used in computer science to model problems that can be represented as networks, such as determining routes in a city or designing circuits. It allows analyzing properties of graphs like connectivity and determining the minimum number of resources required. A graph can be represented using an adjacency matrix or adjacency list to store information about its vertices and edges.
A perfect matching in a bipartite graph is a matching that matches all vertices. The document discusses algorithms for finding a perfect matching in regular bipartite graphs. The currently most efficient algorithm takes time O(m), where m is the number of edges and n is the number of vertices. The document improves this to O(min{m, n2.5ln n/d}) by proving a uniform sampling theorem: sampling each edge independently with probability O(n ln n/d2) results in a graph that has a perfect matching with high probability.
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.
The document discusses graphs and graph algorithms. It begins by defining what a graph is - a collection of vertices connected by edges. It then lists four learning objectives related to representing graphs, traversing graphs, calculating minimum spanning trees, and finding shortest routes. The document goes on to describe different ways of representing graphs through adjacency matrices and lists. It also explains graph traversal algorithms like depth-first search and breadth-first search. Finally, it discusses algorithms for finding minimum spanning trees and shortest paths in weighted graphs.
This document discusses various graph parameters related to distance between vertices in a graph, including diameter, radius, and average distance. It examines how these parameters are defined, how they relate to each other and other graph properties, and how they behave in different graph classes. It also discusses characterizations of graph classes based on distance properties and generalizations to concepts like Steiner trees.
The document discusses graphs and graph algorithms. It defines what a graph is and how they can be represented. It also explains graph traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). Additionally, it covers algorithms for finding the shortest path using Dijkstra's algorithm and calculating minimum spanning trees using Kruskal's algorithm.
This document provides information about graphs and graph algorithms. It discusses different graph representations including adjacency matrices and adjacency lists. It also describes common graph traversal algorithms like depth-first search and breadth-first search. Finally, it covers minimum spanning trees and algorithms to find them, specifically mentioning Kruskal's algorithm.
This document is a project report submitted by S. Manikanta in partial fulfillment of the requirements for a Master of Science degree in Mathematics. The report discusses applications of graph theory. It provides an overview of graph theory concepts such as definitions of graphs, terminology used in graph theory, different types of graphs, trees and forests, graph isomorphism and operations, walks and paths in graphs, representations of graphs using matrices, applications of graphs in areas like computer science, fingerprint recognition, security, and more. The document also includes examples and illustrations to explain various graph theory concepts.
Unit II_Graph.pptxkgjrekjgiojtoiejhgnltegjtepournima055
The document discusses graphs and graph algorithms. It covers basic graph concepts like vertices, edges, paths, cycles. It describes different ways to represent graphs like adjacency matrix, adjacency list and their pros and cons. It also discusses operations on graphs like inserting and deleting vertices and edges. The document explains traversal algorithms like depth-first search and breadth-first search. It covers minimum spanning tree algorithms like Prim's and Kruskal's. It also briefly discusses shortest path algorithms like Dijkstra's and topological sorting.
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.
This document provides an introduction to topological graph theory. It begins with definitions of basic graph theory concepts from a topological perspective, such as representing graphs with curved arcs instead of straight lines. It then discusses graph drawings, incidence matrices, vertex valence, and graph maps/isomorphisms. Important classes of graphs are introduced, such as trees, paths, cycles, and complete graphs. The document aims to introduce preliminary concepts of topological graph theory to students in a simple manner.
The document discusses Hamiltonian graphs, which are graphs that contain a Hamiltonian circuit or path. A Hamiltonian circuit visits each vertex in the graph exactly once and forms a cycle. The document provides an example of a graph that is Hamiltonian but not Eulerian, and vice versa. It notes that while it is clear only connected graphs can be Hamiltonian, there is no simple criterion to determine if a graph is Hamiltonian like there is for Eulerian graphs. It presents Dirac's sufficient condition for a graph to be Hamiltonian.
This document defines and describes different types of graphs. It begins by defining what a graph is as consisting of vertices and edges. It then discusses finite vs infinite graphs, isolated and pendant vertices, null graphs, regular graphs, subgraphs, complete graphs, bipartite graphs, and complete bipartite graphs. Diagrams are provided to illustrate each graph type.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
Lection is one of the course "Discrete Mathematics." More detailed information about the programs of the Economic Cybernetics Department SumDU you could find here: https://ek.biem.sumdu.edu.ua/courses
This document discusses tree data structures and binary search trees. It begins with definitions of trees and their components like nodes, edges, and root/child relationships. It then covers tree traversals like preorder, inorder and postorder. Next, it defines binary search trees and their properties for storing and retrieving data efficiently. It provides examples of inserting, searching for, and deleting nodes from binary search trees. Searching and insertion are straightforward, while deletion has three cases depending on the number of child nodes.
This document discusses regular languages and finite automata. It begins with an overview of regular expressions and the introduction to the theory of finite automata. It then explains that finite automata are used in text processing, compilers, and hardware design. The document goes on to define automata theory and discusses deterministic finite automata (DFA) and nondeterministic finite automata (NFA). It notes that while NFA allow epsilon transitions and multiple state transitions, DFA and NFA have equivalent computational power.
Lection is one of the course "Discrete Mathematics." More detailed information about the programs of the Economic Cybernetics Department SumDU you could find here: https://ek.biem.sumdu.edu.ua/courses
This document outlines the topics to be covered in Lection 2, including propositions and compound statements, basic laws of logical operations, and propositional functions and quantifiers. It provides the plan to discuss propositions and compound statements in section 3.1, basic laws of logical operations in section 3.2, and propositional functions and quantifiers in section 3.3, with an example given for section 3.2.
This document discusses the transportation problem of linear programming. It involves transporting a commodity from sources to destinations with the given shipping costs provided in Table 1. Table 2 shows the constraint coefficients for the transportation problem. The document experiments with the obtained data to find the appropriate values to solve this transportation problem.
Кіберзагрози у фінансовій системі Європи та України.pdfssuser039bf6
Більше інформації по темі «Практики ЄС щодо захисту фінансової системи від кіберзагроз» за посиланням: https://ek.biem.sumdu.edu.ua/erazmus-jean-monnet
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
2. Plan
1. Definition of the planar graph.
2. Kuratowski's and Wagner's theorems.
3. Planarity criteria.
3. 1. Definition of the planar graph.
A planar graph is a graph that can be embedded in
the plane, i.e., it can be drawn on the plane in such a way
that its edges intersect only at their endpoints. In other
words, it can be drawn in such a way that no edges cross
each other. Such a drawing is called a plane
graph or planar embedding of the graph. A plane graph
can be defined as a planar graph with a mapping from
every node to a point on a plane, and from every edge to
a plane curve on that plane, such that the extreme points
of each curve are the points mapped from its end nodes,
and all curves are disjoint except on their extreme points.
4.
5. Plane graphs can be encoded by combinatorial
maps. A combinatorial map is
a combinatorial object
modelling topological structures with subdivided
objects.
6. A Planar graphs
Planar graphs generalize to graphs drawable on a
surface of a given genus. In this terminology, planar
graphs have graph genus 0, since the plane (and the
sphere) are surfaces of genus 0.
genus (plural genera) has a few different, but
closely related, meanings. The most common
concept, the genus of an (orientable) surface, is the
number of "holes" it has, so that a sphere has genus
0 and a torus has genus 1.
8. A TORUS
In geometry, a torus (plural tori) is a surface of
revolution generated by revolving
a circle in three-dimensional space about an axis
that is coplanar with the circle.
9. Genus of orientable surfaces
In simpler terms, the value of an orientable
surface's genus is equal to the number of
"holes" it has.
10. SUMMARIZING
• The genus of a graph is the minimal integer n such that the
graph can be drawn without crossing itself on a sphere
with n handles (i.e. an oriented surface of genus n). Thus,
a planar graph has genus 0, because it can be drawn on a
sphere without self-crossing.
• The non-orientable genus of a graph is the minimal
integer n such that the graph can be drawn without
crossing itself on a sphere with n cross-caps (i.e. a non-
orientable surface of (non-orientable) genus n). (This
number is also called the demigenus.)
• The Euler genus is the minimal integer n such that the
graph can be drawn without crossing itself on a sphere
with n cross-caps or on a sphere with n/2 handles.
11. 2. Kuratowski's and Wagner's theorems
The Polish mathematician Kazimierz Kuratowski provided a
characterization of planar graphs in terms of forbidden graphs, now
known as Kuratowski's theorem:
A finite graph is planar if and only if it does not contain a subgraph that
is a subdivision of the complete graph K5 or the complete bipartite
graph K3,3 (utility graph). A subdivision of a graph results from inserting
vertices into edges (for example, changing an edge •——• to •—•—•)
zero or more times.
An example of a graph with no K5 or K3,3 subgraph. However, it
contains a subdivision of K3,3 and is therefore non-planar.
Instead of considering subdivisions, Wagner's theorem deals
with minors:
A finite graph is planar if and only if it does not have K5 or K3,3 as
a minor.
13. A Forbidden graphs
In graph theory, a branch of mathematics, many
important families of graphs can be described by
a finite set of individual graphs that do not
belong to the family and further exclude all
graphs from the family which contain any of
these forbidden graphs as (induced) subgraph or
minor.
14. A minor of the grafp
In graph theory, an undirected graph H is called
a minor of the graph G if H can be formed
from G by deleting edges and vertices and
by contracting edges.
15. Subdivision and smoothing
In general, a subdivision of a graph G (sometimes
known as an expansion) is a graph resulting from the
subdivision of edges in G.
The subdivision of some edge e with endpoints
{u,v} yields a graph containing one new vertex w, and
with an edge set replacing e by two new edges, {u,w}
and {w,v}. For example, the edge e, with endpoints
{u,v}:
17. Smoothing out
The reverse operation, smoothing
out or smoothing a vertex w with regards to the
pair of edges (e1 ,e2 ) incident on w, removes both
edges containing w and replaces (e1 ,e2 ) with a new
edge that connects the other endpoints of the pair.
Here it is emphasized that only 2-valent
vertices can be smoothed. For example, the
simple connected graph with two edges, e1 {u,w}
and e2 {w,v}:
19. Other planarity criteria
In practice, it is difficult to use Kuratowski's
criterion to quickly decide whether a given
graph is planar. However, there exist
fast algorithms for this problem: for a graph
with n vertices, it is possible to determine in
time O(n) (linear time) whether the graph may
be planar or not (see planarity testing).
20. Other planarity criteria
For a simple, connected, planar graph with v vertices
and e edges and f faces, the following simple conditions hold for v ≥ 3:
• Theorem 1. e ≤ 3v − 6;
• Theorem 2. If there are no cycles of length 3, then e ≤ 2v − 4.
• Theorem 3. f ≤ 2v − 4.
In this sense, planar graphs are sparse graphs, in that they
have only O(v) edges, asymptotically smaller than the maximum O(v2).
The graph K3,3, for example, has 6 vertices, 9 edges, and no cycles of
length 3. Therefore, by Theorem 2, it cannot be planar. These
theorems provide necessary conditions for planarity that are not
sufficient conditions, and therefore can only be used to prove a graph
is not planar, not that it is planar. If both theorem 1 and 2 fail, other
methods may be used.
21. A sparse graph
A dense graph is a graph in which the number of
edges is close to the maximal number of edges.
The opposite, a graph with only a few edges, is
a sparse graph. The distinction between sparse
and dense graphs is rather vague, and depends
on the context.
22. • For undirected simple graphs, the graph
density is defined as:
• For directed simple graphs, the graph density
is defined as:
23. Euler's formula
Euler's formula states that if a finite, connected,
planar graph is drawn in the plane without any
edge intersections, and v is the number of
vertices, e is the number of edges and f is the
number of faces (regions bounded by edges,
including the outer, infinitely large region), then
v-e+f=2.