These definitions cover some basic terms and concepts of graph theory and how they have been implemented as C++ classes.
|
|
- Gerald Weaver
- 5 years ago
- Views:
Transcription
1 hapter 5. raph Theory *raph Theory, *Mathematics raph theory is an area of mathematics which has been incorporated into IS to solve some specific problems in oolean operations and sweeping. It may be also be used to solve other problems in geometric modeling. graph is a mathematical abstraction of relationships. Many real-world situations can conveniently be described through a diagram or graph consisting of a set of points (nodes or vertices) together with lines (edges) joining various pairs of these points. This graphical representation helps us understand connectivity relationships and is the basis for graph theory. In graph diagrams, one is mainly interested in whether or not two given points are joined by a line. The manner in which they are joined long line, short line, straight line, curved line is immaterial, and the relative positions of the vertices and edges have no significance. There is no unique way of drawing a graph. The graph theory subset of IS laws provides a generic way of dealing with finite combinations of objects that have some relation to each other. The graph theory laws deal with the discrete, not continuous, part of mathematics. efinitions *raph Theory These definitions cover some basic terms and concepts of graph theory and how they have been implemented as ++ classes. relation is a set of ordered pairs. If all of the elements of the ordered pairs come from a set S, then the relation may be said to be on the set S. relation R is called symmetric if for each ordered pair (a,b) in R, the ordered pair (b,a) is also in R. graph is a symmetric relation on a set V representing the vertices of S. The ordered pairs of a graph are called edges of the graph. No distinction is made between the pair (a,b) and the pair (b,a). The vertices a and b are said to be adjacent to the edge (a,b). The edge (a,b) is said to be adjacent to the vertices a and b.
2 path is a distinct sequence of vertices v, v 1, v 2,..., v n such that for all i<n, v i is adjacent to v i+1. The vertex v is the start of the path. The vertex v n is the end of the path. The integer n is the length of the path. Sometimes the graph defined by the sequence of vertices together with the edges that connect them is also called a path. n example of a path in igure 5-1 would be the sequence of vertices (,,,, ). graph is called connected if given any two vertices a and b in, there is a path in from a to b. The graph in igure 5-1 is not connected, because there is no path from the vertex to the vertex K. graph S is called a subgraph of if every vertex and edge in S is also in. maximum connected subgraph of is called a component of. igure 5-1 shows a graph with three components, vertices L, and ten edges. igure 5-2 shows a subgraph and one of the three components of the graph depicted in igure 5-1. I J K L igure 5-1. xample of a raph igure 5-2. xample of a Subgraph
3 cycle is a sequence of at vertices v, v 1, v 2,..., v n such that v = v n and v, v 1, v 2,..., v n 1 is a path. graph is called a cycle if it is connected and non empty and if every vertex is of degree two. vertex v is called a cycle vertex of the graph if v belongs to a cycle in the graph. The degree of a vertex is the number of adjacent edges to the vertex. The distance between two vertices in a graph is the length of the shortest path in the graph from one vertex to the other. In igure 5-2, the sequence of vertices (,,,,, ) is a cycle, with each being a cycle vertex. The vertices and are not cycle vertices. The degree of vertices,, and J from igure 5-1 is three. The degree of vertices,, and is two, while the degree of vertices,, I, K, and L is one and the degree of vertex is zero. graph is called a tree if it is connected and does not contain a cycle. graph is called linear if it is a tree and if it does not contain a vertex of degree greater than two. The concepts of vertex, edge, and graph have been implemented as the ++ classes gvertex, gedge, and generic_graph. (entity_gvertex is derived from gvertex except that it contains a pointer to an entity in the model. Such an entity could be a cell or a face.) gvertex may be created with an optional char *name. gedge may be created with two gvertex pointers. n empty graph may be created and edges and vertices may be added to it by calling its add_vertex and add_edge methods. Once created, a graph may be interrogated, ordered, or subsetted in a number of ways. The generic_graph class has methods to tell if a graph is: connected a tree linear a cycle It also has methods to tell how many components the graph has and to return each of the components as a subgraph. The components may also be identified by giving an edge or vertex in them. Real World raph Theory *raph Theory One use of graph theory in geometric modeling is to abstract a given model s cells into a graph. ach cell of the geometric model becomes a point of the graph. Points of the graph are connected with lines (or edges) only if the cells of the geometric model are adjacent with faces. nother use of graph theory is to abstract a given model s faces into a graph. ach face of the geometric model becomes a point of the graph. Points of the graph are connected with lines (or edges) only if the faces of the geometric model are adjacent.
4 In either case, once a graph has been obtained, the graph subsystem can be used to find the shortest path, the shortest cycle, the cut edges and vertices, etc. n example of using graph theory in IS is in selective ooleans and sweeping. ells used as part of sweeping are passed into the graph subsystem. Using graph theory, the shortest path is calculated and used to trim the trees of the graph. Mapping this result back to the original cells (and entities) determines which portions of the resulting swept model to keep and which to throw away. oolean Operations on raphs *raph Theory The graph theory subsystem provides for four types of oolean operations: unite, intersect, subtract and lose boundary, and subtract and keep boundary. igure 5-4 shows the union of graph 1 and graph 2, shown in igure 5-. The result is a graph whose vertices and edges are the union of the vertices and edges of the original graph. s such, there are no duplicate vertices and edges even though they appear in the individual graphs. raph 1 raph 2 igure 5-. raphs 1 and 2 for oolean Operations igure 5-4. raph 1 Union with raph 2
5 igure 5-5 shows the intersection of the two graphs which keeps only those vertices and edges that are common to both. igure 5-5. raph 1 Intersected with raph 2 igure 5-6 and igure 5-7 show the subtraction operations. The result of subtraction can be ambiguous, because an edge (such as ) might be half in a graph. or this reason the subtract method of generic_graph takes a logical option that tells if such edges should be kept or not. igure 5-6 shows graph 2 subtracted from graph 1 with half edges not kept and igure 5-7 show the half edges being kept. igure 5-6. raph 2 Subtracted from raph 1 Not Keeping oundary
6 igure 5-7. raph 2 Subtracted from raph 1 Keeping oundary Types of dges and Vertices *raph Theory vertex v is called a cut vertex of the graph if removing the vertex v and the boundary edges from results in more components than. n edge e is called a cut edge of the graph if removing the edge e from results in more components than. When an edge is removed from a graph, its vertices are left in. Vertices,,, and are cut vertices of the graph in igure 5-8, because removing them from the graph results in more components. I igure 5-8. ut Vertices, ut dges, and ycle Vertices igure 5-9 top shows what happens when vertex is cut from the graph, resulting in two components. Likewise, edges - and - are cut edges of the graph. igure 5-9 bottom shows what happens when edge is cut from the graph, resulting in two components. Vertices,,,,,,, and I are cycle vertices of the graph depicted in igure 5-8.
7 I I igure 5-9. Resulting raphs Minus a ut Vertex and a ut dge lass methods from generic_graph associated with cut vertices and edges include cut_vertices, is_cut_vertex, cut_edges, and is_cut_edge. Scheme extensions include graph:cut vertices, graph:cut_vertex?, graph:cut edges, and graph:cut_edge?. Ordering raphs *raph Theory Ordering a graph in some manner is one of the more fundamental graph operations that yields a wealth of important relationship information. One way is to order the vertices by the distance, or hops, that they are from a given vertex. The distance is not the length of the lines drawn for the edges, but the number of vertices encountered when tracing along the edges from one vertex to another specified vertex. The distance between two vertices in a graph is the number of vertices of the shortest path in the graph from one vertex to the other. The order from method of ordering a graph works well for trees and linear graphs. The two graphs show in igure 5-1 have been ordered by distance from vertex. ecause ordering uses a single vertex as the starting point, it is possible to have multiple nodes in a graph that are the same distance away, as is shown with,, and, or and.
8 igure 5-1. Ordering raphs If a graph is cyclic, then it may be ordered by the order_cyclic method. This sets a given vertex s order to zero and the other vertices in a clockwise cyclic order as shown in igure igure yclic Ordering oth order_cyclic and order from ordered graphs may be negated, and they negate in different ways. igure 5-12 shows the negations of the graphs from igure 5-1 and igure ecause ordering uses a single vertex as the starting point, it is possible to have multiple nodes in a negated ordered graph that are, as is shown with and.
9 igure Negation of the Ordering in raphs nother way to order a graph is to order it with respect to an ordered graph such that is a subgraph of. The order_with method imposes the order of onto and rescales the ordering on to remove gaps. The type of ordering (i.e. cyclic or not) is inherited from the ordered graph. igure 5-1 shows a linear graph imposing its order on a subgraph. raph Subgraph 1 2 igure 5-1. Ordering a raph with another raph igure 5-14 shows a cyclic graph imposing its order on a subgraph.
10 igure Ordering raphs Once a graph has been ordered, the order of a vertex may be found by calling the get_order method and the maximum order in the graph may be found by calling the max_order method. iven an ordered graph, a subgraph may be formed by calling either of the two subset methods of the generic_graph class. One method takes in two integers and the other takes a law pointer. The subset method with two integers takes a and b and returns a subgraph in one of two ways. If a<b, then the set of all vertices with orders between a and b is returned along with all edges that have both of their adjacent vertices in this set. If b<a, then the set of all vertices with orders not between a and b is returned along with all edges that have both of their adjacent vertices in this set. The subset method with a law returns the set of all vertices such that their order evaluates as true along with the all edges that have both of their adjacent vertices evaluating as true orders. In igure 5-15, the graph on the left has been ordered and subsetted from 1 to, resulting in the graph on the right.
11 igure Ordered and Subsetted from 1 to In igure 5-16, the graph on the left has been ordered and subsetted from 4 to 1, resulting in the graph on the right. Note that this is a cyclic graph in ascending order. ence, the result is (,,, ) instead of (,,, ) igure Ordering raphs In igure 5-17, the graph on the top has been subsetted with the law even?(x) which returns true if x is even. In addition to even?, the laws odd?, int?, and prime? are available.
12 1 2 4 igure Ordering a raph with another raph lass methods from generic_graph associated with ordering include subset, order_from, order_cyclic, order_with, negate, get_order, and max_order; while Scheme extensions include graph:get order, graph:order cyclic, graph:order with, graph:order from, graph:show order, and graph:negate. Other Ways to reate raphs *raph Theory method has also been added to return a subgraph that is a shortest path between two vertices in the same component of a graph. The shortest path from to in igure 5-18 is the path (,,,, ). or cycle vertices, a method has been added to return a subgraph that is a shortest cycle that contains the vertex. The shortest cycle of the graph in igure 5-18 that contains is the cycle (,,,, ). igure The onnectivity raph igure 5-19 shows a blank body consisting of a block with two slots cut out of it and a tool body consisting of a cylinder that penetrates the block three times.
13 igure The lank and Tool odies The first stage of the selective ooleans creates a non-regular unite of the two bodies and attaches cellular topology to the result (igure 5-2). In addition, the connectivity graph (igure 5-18) of the resulting cells is returned and the vertices, or cells, of the graph are marked in a way that lets one know which body or bodies they came from. igure 5-2. The lank ody fter the irst Stage of Selective ooleans raphs may be copied by calling the copy method or created for an NTITY_LIST of LLs by calling api_create_graph_from_cells. igure 5-18 is the graph that is created if the eight cells in igure 5-2 are sent to api_create_graph_from_cells. The gvertex and gedge classes also have pointers for holding true or false information as to what kind of vertex or edge they are. The meaning of kind is left to the user. vertex or edge may (or may not) be any number of kinds. or example, the cells of the body in igure 5-2 may be marked as kind (meaning being from the tool body) or kind 1 (meaning being from the blank body). Some cells may be of both kind and kind 1.
14 The kind method of the class generic_graph returns a subgraph of all edges and vertices of, or not of, a given kind. The graph in igure 5-18 is returned by the first stage of the selective oolean on the bodies in igure The linear subgraph formed by the vertices (,,,,,, ) are from the tool body and marked as kind. The subgraph formed by the vertices (,,, ) are from the blank body and are marked as kind 1. The second stage of the selective oolean engine takes the non-regular united body and a list or graph of cells. It returns a regularized body that is made up of the given cells or vertices of the graph (igure 5-21). igure The Result of Picking,, and to Keep ttaching ata to dges and Vertices *raph Theory The ++ classes of gvertex and gedge are use counted in the same way that laws are use counted. That is, they are copied by calling the add method and deleted by calling the remove method. Several graphs may refer to the same edges and vertices without having to copy the data that they may contain. To make a gvertex or gedge contain data, derive a class from the base classes of gvertex and gedge. Use a technique similar to the entity_gvertex class which enables it to contain an entity pointer. ase methods may be added to locate vertices by name or by entity, so that type casting is not necessary. method can also be added to return all the entities referenced by a graph.
Classes Fa thru Kz. Chapter 32. FACE
Chapter 32. Classes Fa thru Kz Topic: Ignore FACE Class: Purpose: Model Topology, SAT Save and Restore Represents a bounded portion of a SURFACE. Derivation: FACE : ENTITY : ACIS_OBJECT : SAT Identifier:
More informationChapter 9: Elementary Graph Algorithms Basic Graph Concepts
hapter 9: Elementary Graph lgorithms asic Graph oncepts msc 250 Intro to lgorithms graph is a mathematical object that is used to model different situations objects and processes: Linked list Tree (partial
More informationChapter 8 Topics in Graph Theory
Chapter 8 Topics in Graph Theory Chapter 8: Topics in Graph Theory Section 8.1: Examples {1,2,3} Section 8.2: Examples {1,2,4} Section 8.3: Examples {1} 8.1 Graphs Graph A graph G consists of a finite
More informationDesign and Analysis of Algorithms
S 101, Winter 2018 esign and nalysis of lgorithms Lecture 3: onnected omponents lass URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ nd of Lecture 2: s, Topological Order irected cyclic raph.g., Vertices
More informationAnalysis of Algorithms Prof. Karen Daniels
UMass Lowell omputer Science 91.404 nalysis of lgorithms Prof. Karen aniels Spring, 2013 hapter 22: raph lgorithms & rief Introduction to Shortest Paths [Source: ormen et al. textbook except where noted]
More informationGeometric Modeling. Introduction
Geometric Modeling Introduction Geometric modeling is as important to CAD as governing equilibrium equations to classical engineering fields as mechanics and thermal fluids. intelligent decision on the
More informationLesson 22: Basic Graph Concepts
Lesson 22: asic Graph oncepts msc 175 iscrete Mathematics 1. Introduction graph is a mathematical object that is used to model different relations between objects and processes: Linked list Flowchart of
More information5 Graphs
5 Graphs jacques@ucsd.edu Some of the putnam problems are to do with graphs. They do not assume more than a basic familiarity with the definitions and terminology of graph theory. 5.1 Basic definitions
More informationTaibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103. Chapter 2. Sets
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H.
More information3.1 Basic Definitions and Applications
Graphs hapter hapter Graphs. Basic efinitions and Applications Graph. G = (V, ) n V = nodes. n = edges between pairs of nodes. n aptures pairwise relationship between objects: Undirected graph represents
More informationNumber System. Introduction. Natural Numbers (N) Whole Numbers (W) Integers (Z) Prime Numbers (P) Face Value. Place Value
1 Number System Introduction In this chapter, we will study about the number system and number line. We will also learn about the four fundamental operations on whole numbers and their properties. Natural
More informationFundamental Properties of Graphs
Chapter three In many real-life situations we need to know how robust a graph that represents a certain network is, how edges or vertices can be removed without completely destroying the overall connectivity,
More informationA graph is finite if its vertex set and edge set are finite. We call a graph with just one vertex trivial and all other graphs nontrivial.
2301-670 Graph theory 1.1 What is a graph? 1 st semester 2550 1 1.1. What is a graph? 1.1.2. Definition. A graph G is a triple (V(G), E(G), ψ G ) consisting of V(G) of vertices, a set E(G), disjoint from
More informationTCOM 501: Networking Theory & Fundamentals. Lecture 11 April 16, 2003 Prof. Yannis A. Korilis
TOM 50: Networking Theory & undamentals Lecture pril 6, 2003 Prof. Yannis. Korilis 2 Topics Routing in ata Network Graph Representation of a Network Undirected Graphs Spanning Trees and Minimum Weight
More informationGraph Theory. Connectivity, Coloring, Matching. Arjun Suresh 1. 1 GATE Overflow
Graph Theory Connectivity, Coloring, Matching Arjun Suresh 1 1 GATE Overflow GO Classroom, August 2018 Thanks to Subarna/Sukanya Das for wonderful figures Arjun, Suresh (GO) Graph Theory GATE 2019 1 /
More information1. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which:
P R O B L E M S Finite Autom ata. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which: a) Are a multiple of three in length. b) End with the string
More informationGeometric Constructions
HISTORY OF MATHEMATICS Spring 2005 Geometric Constructions Notes, activities, assignment; #3 in a series. Note: I m not giving a specific due date for this somewhat vague assignment. The idea is that it
More informationLesson 5.5. Minimum Spanning Trees. Explore This
Lesson. St harles ounty Minimum Spanning Trees s in many of the previous lessons, this lesson focuses on optimization. Problems and applications here center on two types of problems: finding ways of connecting
More informationChapter 12 Solid Modeling. Disadvantages of wireframe representations
Chapter 12 Solid Modeling Wireframe, surface, solid modeling Solid modeling gives a complete and unambiguous definition of an object, describing not only the shape of the boundaries but also the object
More information2.2 Set Operations. Introduction DEFINITION 1. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, EXAMPLE 2
2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of
More informationL10 Graphs. Alice E. Fischer. April Alice E. Fischer L10 Graphs... 1/37 April / 37
L10 Graphs lice. Fischer pril 2016 lice. Fischer L10 Graphs... 1/37 pril 2016 1 / 37 Outline 1 Graphs efinition Graph pplications Graph Representations 2 Graph Implementation 3 Graph lgorithms Sorting
More informationEulerian Cycle (2A) Young Won Lim 5/11/18
ulerian ycle (2) opyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the NU ree ocumentation License, Version 1.2 or any later
More informationRelational Database: The Relational Data Model; Operations on Database Relations
Relational Database: The Relational Data Model; Operations on Database Relations Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Overview
More informationCHAPTER-2 A GRAPH BASED APPROACH TO FIND CANDIDATE KEYS IN A RELATIONAL DATABASE SCHEME **
HPTR- GRPH S PPROH TO FIN NIT KYS IN RLTIONL TS SHM **. INTROUTION Graphs have many applications in the field of knowledge and data engineering. There are many graph based approaches used in database management
More informationA GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY KARL L. STRATOS Abstract. The conventional method of describing a graph as a pair (V, E), where V and E repectively denote the sets of vertices and edges,
More informationBoolean Component. Chapter 1. Boolean Operations
Chapter 1. Boolean Component Component: The Boolean Component (BOOL), in the bool directory, performs Boolean operations on the model topology of bodies, first finding the intersections between bodies,
More informationA set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N
Mathematical Preliminaries Read pages 529-540 1. Set Theory 1.1 What is a set? A set is a collection of entities of any kind. It can be finite or infinite. A = {a, b, c} N = {1, 2, 3, } An entity is an
More informationGRAPH THEORY. What are graphs? Why graphs? Graphs are usually used to represent different elements that are somehow related to each other.
GRPH THEORY Hadrian ng, Kyle See, March 2017 What are graphs? graph G is a pair G = (V, E) where V is a nonempty set of vertices and E is a set of edges e such that e = {a, b where a and b are vertices.
More informationBayesian Networks and Decision Graphs
ayesian Networks and ecision raphs hapter 7 hapter 7 p. 1/27 Learning the structure of a ayesian network We have: complete database of cases over a set of variables. We want: ayesian network structure
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More information1. Introduction to Constructive Solid Geometry (CSG)
opyright@010, YZU Optimal Design Laboratory. All rights reserved. Last updated: Yeh-Liang Hsu (010-1-10). Note: This is the course material for ME550 Geometric modeling and computer graphics, Yuan Ze University.
More informationSlides for Faculty Oxford University Press All rights reserved.
Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,
More informationSETS. Sets are of two sorts: finite infinite A system of sets is a set, whose elements are again sets.
SETS A set is a file of objects which have at least one property in common. The objects of the set are called elements. Sets are notated with capital letters K, Z, N, etc., the elements are a, b, c, d,
More informationCurves and Surfaces. Chapter 7. Curves. ACIS supports these general types of curves:
Chapter 7. Curves and Surfaces This chapter discusses the types of curves and surfaces supported in ACIS and the classes used to implement them. Curves ACIS supports these general types of curves: Analytic
More informationSolid Modeling Lecture Series. Prof. Gary Wang Department of Mechanical and Manufacturing Engineering The University of Manitoba
Solid Modeling 25.353 Lecture Series Prof. Gary Wang Department of Mechanical and Manufacturing Engineering The University of Manitoba Information complete, unambiguous, accurate solid model Solid Modeling
More informationNote. Out of town Thursday afternoon. Willing to meet before 1pm, me if you want to meet then so I know to be in my office
raphs and Trees Note Out of town Thursday afternoon Willing to meet before pm, email me if you want to meet then so I know to be in my office few extra remarks about recursion If you can write it recursively
More informationGeometry Unit 4a - Notes Triangle Relationships
Geometry Unit 4a - Notes Triangle Relationships This unit is broken into two parts, 4a & 4b. test should be given following each part. Triangle - a figure formed by three segments joining three noncollinear
More informationNetwork Topology and Graph
Network Topology Network Topology and Graph EEE442 Computer Method in Power System Analysis Any lumped network obeys 3 basic laws KVL KCL linear algebraic constraints Ohm s law Anawach Sangswang Dept.
More informationGeometric Modeling Mortenson Chapter 11. Complex Model Construction
Geometric Modeling 91.580.201 Mortenson Chapter 11 Complex Model Construction Topics Topology of Models Connectivity and other intrinsic properties Graph-Based Models Emphasize topological structure Boolean
More informationPolygon Meshes and Implicit Surfaces
CSCI 420 Computer Graphics Lecture 9 Polygon Meshes and Implicit Surfaces Polygon Meshes Implicit Surfaces Constructive Solid Geometry [Angel Ch. 10] Jernej Barbic University of Southern California 1 Modeling
More informationPolygon Meshes and Implicit Surfaces
CSCI 420 Computer Graphics Lecture 9 and Constructive Solid Geometry [Angel Ch. 10] Jernej Barbic University of Southern California Modeling Complex Shapes An equation for a sphere is possible, but how
More informationLesson 4.2. Critical Paths
Lesson. ritical Paths It is relatively easy to find the shortest time needed to complete a project if the project consists of only a few activities. ut as the tasks increase in number, the problem becomes
More informationChapter 3. Set Theory. 3.1 What is a Set?
Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or described. Any
More informationComputer Aided Engineering Design Prof. Anupam Saxena Department of Mechanical Engineering Indian Institute of Technology, Kanpur.
(Refer Slide Time: 00:28) Computer Aided Engineering Design Prof. Anupam Saxena Department of Mechanical Engineering Indian Institute of Technology, Kanpur Lecture - 6 Hello, this is lecture number 6 of
More informationSAMPLE. MODULE 5 Undirected graphs
H P T R MOUL Undirected graphs How do we represent a graph by a diagram and by a matrix representation? How do we define each of the following: graph subgraph vertex edge (node) loop isolated vertex bipartite
More informationDiscrete Mathematics
Discrete Mathematics Lecture 2: Basic Structures: Set Theory MING GAO DaSE@ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 18, 2017 Outline 1 Set Concepts 2 Set Operations 3 Application
More informationA.1 Numbers, Sets and Arithmetic
522 APPENDIX A. MATHEMATICS FOUNDATIONS A.1 Numbers, Sets and Arithmetic Numbers started as a conceptual way to quantify count objects. Later, numbers were used to measure quantities that were extensive,
More informationA region is each individual area or separate piece of the plane that is divided up by the network.
Math 135 Networks and graphs Key terms Vertex (Vertices) ach point of a graph dge n edge is a segment that connects two vertices. Region region is each individual area or separate piece of the plane that
More informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More informationThe clique number of a random graph in (,1 2) Let ( ) # -subgraphs in = 2 =: ( ) We will be interested in s.t. ( )~1. To gain some intuition note ( )
The clique number of a random graph in (,1 2) Let () # -subgraphs in = 2 =:() We will be interested in s.t. ()~1. To gain some intuition note ()~ 2 =2 and so ~2log. Now let us work rigorously. () (+1)
More informationDomination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes
Domination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes Leonor Aquino-Ruivivar Mathematics Department, De La Salle University Leonorruivivar@dlsueduph
More informationThe clique number of a random graph in (,1 2) Let ( ) # -subgraphs in = 2 =: ( ) 2 ( ) ( )
1 The clique number of a random graph in (,1 2) Let () # -subgraphs in = 2 =:() We will be interested in s.t. ()~1. To gain some intuition note ()~ 2 =2 and so ~2log. Now let us work rigorously. () (+1)
More informationGEOMETRY BASIC GEOMETRICAL IDEAS. 3) A point has no dimensions (length, breadth or thickness).
CLASS 6 - GEOMETRY BASIC GEOMETRICAL IDEAS Geo means Earth and metron means Measurement. POINT 1) The most basic shape in geometry is the Point. 2) A point determines a location. 3) A point has no dimensions
More informationChapter 6. Planar Orientations. 6.1 Numberings of Digraphs
Chapter 6 Planar Orientations In this chapter we will focus on algorithms and techniques used for drawing planar graphs. The algorithms we will use are based on numbering the vertices and orienting the
More informationGeometric Entities for Pilot3D. Copyright 2001 by New Wave Systems, Inc. All Rights Reserved
Geometric Entities for Pilot3D Copyright 2001 by New Wave Systems, Inc. All Rights Reserved Introduction on Geometric Entities for Pilot3D The best way to develop a good understanding of any Computer-Aided
More informationSpanning Trees. CSE373: Data Structures & Algorithms Lecture 17: Minimum Spanning Trees. Motivation. Observations. Spanning tree via DFS
Spanning Trees S: ata Structures & lgorithms Lecture : Minimum Spanning Trees simple problem: iven a connected undirected graph =(V,), find a minimal subset of edges such that is still connected graph
More information11/1/13. Polygon Meshes and Implicit Surfaces. Shape Representations. Polygon Models in OpenGL. Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 7 and Constructive Solid Geometry [Angel Ch. 12.1-12.3] Jernej Barbic University of Southern California Modeling Complex Shapes An equation for a sphere is possible,
More informationVertex-Based (Lath) Representations for Three-Dimensional Objects and Meshes p.1/15
Vertex-ased (Lath) Representations for Three-imensional Objects and Meshes Hanan Samet hjs@cs.umd.edu www.cs.umd.edu/ hjs epartment of omputer Science enter for utomation Research Institute for dvanced
More informationAlgorithm Design (8) Graph Algorithms 1/2
Graph Algorithm Design (8) Graph Algorithms / Graph:, : A finite set of vertices (or nodes) : A finite set of edges (or arcs or branches) each of which connect two vertices Takashi Chikayama School of
More informationGraph Theory S 1 I 2 I 1 S 2 I 1 I 2
Graph Theory S I I S S I I S Graphs Definition A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) ( ) V (G). x and y are the endpoints of edge e = {x, y}. They are called adjacent
More informationThe Three Dimensional Coordinate System
The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the
More informationDepth-First Search A B D E C
epth-first Search Outline and Reading efinitions (6.1) Subgraph onnectivity Spanning trees and forests epth-first search (6.3.1) lgorithm xample Properties nalysis pplications of FS (6.5) Path finding
More information9/19/12. Why Study Discrete Math? What is discrete? Sets (Rosen, Chapter 2) can be described by discrete math TOPICS
What is discrete? Sets (Rosen, Chapter 2) TOPICS Discrete math Set Definition Set Operations Tuples Consisting of distinct or unconnected elements, not continuous (calculus) Helps us in Computer Science
More informationCS100: DISCRETE STRUCTURES
CS: DISCRETE STRUCTURES Computer Science Department Lecture : Set and Sets Operations (Ch2) Lecture Contents 2 Sets Definition. Some Important Sets. Notation used to describe membership in sets. How to
More informationUnit 1, Lesson 1: Moving in the Plane
Unit 1, Lesson 1: Moving in the Plane Let s describe ways figures can move in the plane. 1.1: Which One Doesn t Belong: Diagrams Which one doesn t belong? 1.2: Triangle Square Dance m.openup.org/1/8-1-1-2
More informationThe Graphs of Triangulations of Polygons
The Graphs of Triangulations of Polygons Matthew O Meara Research Experience for Undergraduates Summer 006 Basic Considerations Let Γ(n) be the graph with vertices being the labeled planar triangulation
More informationCS302 - Data Structures using C++
CS302 - Data Structures using C++ Topic: Graphs - Introduction Kostas Alexis Terminology In the context of our course, graphs represent relations among data items G = {V,E} A graph is a set of vertices
More informationFault Tree Analysis (Minimal Cutset)
Fault Tree Analysis (Minimal Cutset) Arshad Ahmad arshad@utm.my asic Set Theory 2 Sets Definition. A Set is any well defined collection of objects. Definition. The elements of a set are the objects in
More informationSection 3.1: Nonseparable Graphs Cut vertex of a connected graph G: A vertex x G such that G x is not connected. Theorem 3.1, p. 57: Every connected
Section 3.1: Nonseparable Graphs Cut vertex of a connected graph G: A vertex x G such that G x is not connected. Theorem 3.1, p. 57: Every connected graph G with at least 2 vertices contains at least 2
More informationDHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI
DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI Department of Computer Science and Engineering CS6702 - GRAPH THEORY AND APPLICATIONS Anna University 2 & 16 Mark Questions & Answers Year / Semester: IV /
More informationDefinition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1
Graph fundamentals Bipartite graph characterization Lemma. If a graph contains an odd closed walk, then it contains an odd cycle. Proof strategy: Consider a shortest closed odd walk W. If W is not a cycle,
More informationExtracting consistent and manifold interfaces from multi-valued volume data sets
Extracting consistent and manifold interfaces from multi-valued volume data sets Stephan Bischoff, Leif Kobbelt Lehrstuhl für Informatik 8, RWTH Aachen, 52056 Aachen Email: {bischoff,kobbelt}@informatik.rwth-aachen.de
More informationGraph Theory: Introduction
Graph Theory: Introduction Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in Resources Copies of slides available at: http://www.facweb.iitkgp.ernet.in/~pallab
More informationPlanar Graphs. 1 Graphs and maps. 1.1 Planarity and duality
Planar Graphs In the first half of this book, we consider mostly planar graphs and their geometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter
More information4.4 3D Shape Interpolation - changing one 3D object into another
4.4 3D Shape Interpolation - changing one 3D object into another 1 Two categories: surface based approach (2 nd case) volume based approach (1 st case) Turk/O Brien 2 Surface based approach (2nd case)
More informationA simple problem that has a solution that is far deeper than expected!
The Water, Gas, Electricity Problem A simple problem that has a solution that is far deeper than expected! Consider the diagram below of three houses and three utilities: water, gas, and electricity. Each
More informationSet and Set Operations
Set and Set Operations Introduction A set is a collection of objects. The objects in a set are called elements of the set. A well defined set is a set in which we know for sure if an element belongs to
More informationCSE 332: Data Structures & Parallelism Lecture 19: Introduction to Graphs. Ruth Anderson Autumn 2018
SE 332: ata Structures & Parallelism Lecture 19: Introduction to Graphs Ruth nderson utumn 2018 Today Graphs Intro & efinitions 11/19/2018 2 Graphs graph is a formalism for representing relationships among
More informationLesson 4: Surface Re-limitation and Connection
Lesson 4: Surface Re-limitation and Connection In this lesson you will learn how to limit the surfaces and form connection between the surfaces. Lesson contents: Case Study: Surface Re-limitation and Connection
More informationChapter 4. Relations & Graphs. 4.1 Relations. Exercises For each of the relations specified below:
Chapter 4 Relations & Graphs 4.1 Relations Definition: Let A and B be sets. A relation from A to B is a subset of A B. When we have a relation from A to A we often call it a relation on A. When we have
More informationPerimeter Magic Polygons
Perimeter Magic Polygons In, Terrel Trotter, Jr., then a math teacher in Urbana Illinois, published an article called Magic Triangles of Order n. In, he published a follow up article called Perimeter Magic
More informationSymmetric Product Graphs
Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 5-20-2015 Symmetric Product Graphs Evan Witz Follow this and additional works at: http://scholarworks.rit.edu/theses
More informationGenerating All Simple Convexly-Drawable Polar Symmetric 6-Venn Diagrams
Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn Diagrams Khalegh Mamakani and Frank Ruskey Dept. of Computer Science, University of Victoria, Canada. Abstract. An n-venn diagram consists
More informationHamiltonian cycles in bipartite quadrangulations on the torus
Hamiltonian cycles in bipartite quadrangulations on the torus Atsuhiro Nakamoto and Kenta Ozeki Abstract In this paper, we shall prove that every bipartite quadrangulation G on the torus admits a simple
More informationGraphs: Topological Sort / Graph Traversals (Chapter 9)
Today s Outline raphs: Topological Sort / raph Traversals (hapter 9) S 373 ata Structures and lgorithms dmin: omework #4 - due Thurs, Nov 8 th at pm Midterm 2, ri Nov 6 raphs Representations Topological
More informationBinary Relations McGraw-Hill Education
Binary Relations A binary relation R from a set A to a set B is a subset of A X B Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B. We can also represent
More informationMath 222 (A1) Solutions to Assignment 4
Math (1) Solutions to ssignment 1. We have captured several people whom we suspect are part of a spy ring. They are identified as,,,,,, and. fter interrogation, admits to having met the other six. admits
More informationCS S Lecture February 13, 2017
CS 6301.008.18S Lecture February 13, 2017 Main topics are #Voronoi-diagrams, #Fortune. Quick Note about Planar Point Location Last week, I started giving a difficult analysis of the planar point location
More information27 Geometric Intersection
27 eometric ntersection ------------------------------------------------------------- A natural problem that arises frequently in applications involving geometric data is: "iven a set of N objects, do
More informationTwo Characterizations of Hypercubes
Two Characterizations of Hypercubes Juhani Nieminen, Matti Peltola and Pasi Ruotsalainen Department of Mathematics, University of Oulu University of Oulu, Faculty of Technology, Mathematics Division, P.O.
More informationReview of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16
Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special
More informationKey Graph Theory Theorems
Key Graph Theory Theorems Rajesh Kumar MATH 239 Intro to Combinatorics August 19, 2008 3.3 Binary Trees 3.3.1 Problem (p.82) Determine the number, t n, of binary trees with n edges. The number of binary
More informationModule 5 Graph Algorithms
Module 5 Graph lgorithms Dr. Natarajan Meghanathan Professor of Computer Science Jackson State University Jackson, MS 97 E-mail: natarajan.meghanathan@jsums.edu 5. Graph Traversal lgorithms Depth First
More informationMath 202 Test Problem Solving, Sets, and Whole Numbers 19 September, 2008
Math 202 Test Problem Solving, Sets, and Whole Numbers 19 September, 2008 Ten questions, each worth the same amount. Complete six of your choice. I will only grade the first six I see. Make sure your name
More informationNP-Complete Problems
NP-omplete Problems P and NP Polynomial time reductions Satisfiability Problem, lique Problem, Vertex over, and ominating Set 10/19/2009 SE 5311 FLL 2009 KUMR 1 Polynomial lgorithms Problems encountered
More informationMathematics and Symmetry: A Bridge to Understanding
Mathematics and Symmetry: A Bridge to Understanding Gail Kaplan Department of Mathematics Towson University Towson, Maryland 5, USA gkaplan@towson.edu Abstract This paper describes how to teach group theory
More informationLecture 5: More Examples/Applications of Quasirandom Graphs
Random and Quasirandom Graphs Instructor: Padraic Bartlett Lecture 5: More Examples/Applications of Quasirandom Graphs Week 4 Mathcamp 2012 In our last class, we saw that the Paley graphs were an example
More informationCurriki Geometry Glossary
Curriki Geometry Glossary The following terms are used throughout the Curriki Geometry projects and represent the core vocabulary and concepts that students should know to meet Common Core State Standards.
More informationCPS222 Lecture: Sets. 1. Projectable of random maze creation example 2. Handout of union/find code from program that does this
CPS222 Lecture: Sets Objectives: last revised April 16, 2015 1. To introduce representations for sets that can be used for various problems a. Array or list of members b. Map-based representation c. Bit
More informationDominating Sets in Triangulations on Surfaces
Dominating Sets in Triangulations on Surfaces Hong Liu Department of Mathematics University of Illinois This is a joint work with Michael Pelsmajer May 14, 2011 Introduction A dominating set D V of a graph
More information