Module 2: NETWORKS AND DECISION MATHEMATICS


 Imogene McCoy
 1 years ago
 Views:
Transcription
1 Further Mathematics 2017 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks Key knowledge the conventions, terminology, properties and types of graphs; edge, face, loop, vertex, the degree of a vertex, isomorphic and connected graphs, and the adjacency matrix, and Euler s formula for planar graphs and its application the exploring and travelling problem, walks, trails, paths, Eulerian trails and circuits, and Hamiltonian cycles the minimum connector problem, trees, spanning trees and minimum spanning trees the shortest path problem and Dijkstra s algorithm Key skills construct graphs, digraphs and networks and their matrix equivalents to model and analyse practical situations recognise that a problem is an example of the exploring and travelling problem and to solve it by utilising the concepts of walks, trails, paths, Eulerian trails and circuits, and Hamiltonian paths and cycles recognise that a problem is an example of the minimum connector problem and solve it by utilising the properties of trees, spanning trees and by determining a minimum spanning tree by inspection or using Prim s algorithm for larger scale problems recognise that a problem is an example of the shortest path problem and solve it by inspection or using Dijkstra s algorithm for larger scale problems Chapter Sections Questions to be completed 9.2 Basic concepts 2, 4, 6, 8, 10, 12, 14, 15ab, 16a, 17, 20, 21, Planar graphs and Euler s formula 1, 2, 4, 5, 6, 9ac, 10, 12, 14, 15, 17, Walks, trails, paths, cycles and circuits 1, 2, 3, 4, 5, 6, 8, 9, 13, 17, 18, 19, 20, Trees and their application 1, 2, 7, 8, 10, 13, 15, 16, 17, 18, 21ac, 22a, 23, 24
2 MODULE 2: NETWORKS AND DECISION MATHEMATICS Table of Contents CHAPTER 9 UNDIRECTED GRAPHS AND NETWORKS... 1 KEY KNOWLEDGE... 1 KEY SKILLS BASIC CONCEPTS OF A NETWORK... 3 DEFINITION OF A NETWORK... 3 WORKED EXAMPLE THE DEGREE OF A VERTEX... 3 WORKED EXAMPLE REPRESENTATION OF NETWORKS... 4 WORKED EXAMPLE WORKED EXAMPLE MATRIX REPRESENTATION OF NETWORKS... 5 WORKED EXAMPLE PLANAR, GRAPHS AND EULER S FORMULA... 6 DEGENERATE GRAPH... 6 COMPLETE GRAPH... 6 PLANAR GRAPHS... 6 THE REGIONS OF A PLANAR GRAPH... 7 CONVERTING A NON PLANAR GRAPH... 7 WORKED EXAMPLE WORKED EXAMPLE CONVERTING THREE DIMENSIONAL SOLIDS TO PLANAR GRAPHS... 8 WORKED EXAMPLE EULER S FORMULA... 8 WORKED EXAMPLE TREES AND THEIR APPLICATION GRAPHS AND SUBGRAPHS TREES WORKED EXAMPLE SHORTEST PATHS WORKED EXAMPLE A SHORTEST PATH ALGORITHM WORKED EXAMPLE SPANNING TREES MINIMUM SPANNING TREE AND PRIM S ALGORITHM PRIM S ALGORITHM WORKED EXAMPLE MAXIMUM SPANNING TREE WORKED EXAMPLE DIJKSTRA S ALGORITHM DIJKSTRA S ALGORITHM EXAMPLE DIJKSTRA S ALGORITHM EXAMPLE WORKED EXAMPLE 20 (STUDENT PRACTICE) ADDITIONAL DIJKSTRA S ALGORITHM QUESTIONS WALKS, TRAILS, PATHS, CYCLES AND CIRCUITS... 9 WALKS... 9 CONNECTED GRAPHS... 9 EULER TRAILS... 9 WORKED EXAMPLE EULER CIRCUITS WORKED EXAMPLE AN EULER CIRCUIT ALGORITHM WORKED EXAMPLE PATH AND CYCLES WORKED EXAMPLE HAMILTONIAN CYCLES WORKED EXAMPLE Page 2 of 22
3 9.2 Basic concepts of a network UNDIRECTED GRAPHS AND NETWORKS Definition of a network The telephone system, your family tree and the internet are all considered to be networks (graphs). The simplest definition of a network is: Consider the following diagram. It is perhaps the simplest network. It consists of two vertices (singular vertex) (circles labelled 1 and 2) and one connection between them. This connection is called an edge. Worked Example 1 Count the number of vertices and edges in the network shown. Vertices: Edges: There are two things worth noting about this classification of a network: 1. the vertices and edges can be labelled in any order, using any suitable labelling system. 2. vertices may have different number of edges connected to them. The degree of a vertex Each vertex may have a number of edges connecting it with the rest of the network. This number is called the degree. A vertex with degree 0 is not connected to any other vertex, and is called an isolated vertex. If two (or more) edges connect the same pair of vertices they are called parallel edges (or multiple edges) and all count towards the degree. If there is only one connection between two vertices, the connection is called a simple, or single, connection. Page 3 of 22
4 MODULE 2: NETWORKS AND DECISION MATHEMATICS Worked Example 2 Determine the degree of each vertex in the figure shown. Representation of networks So far we have seen graphical representation of networks. Two other ways of representing networks are: 1. labelling the vertices and labelling the edges according to their vertices 2. matrix representation Worked Example 3 Label the vertices and edges for the figure shown, as in Worked example 2 Page 4 of 22
5 Worked Example 4 Construct a graph from the following list of vertices and edges. V = {A, B, C, D, E} E = {(A, B), (A, C), (A, D), (B, C), (B, D), (B, D), (C, E), (D, E), (E, E)} UNDIRECTED GRAPHS AND NETWORKS There may be other geometric configurations which can be drawn from the same vertex and edge list, but they isomorphic (or equivalent) to this one. Matrix representation of networks A method of representing a network is through the use of a matrix. Worked Example 5 Represent the network shown as a matrix In a matrix representation 1. the sum of a row (or column) gives a degree of that vertex, except where a loop is present (shown as a 1 in the leading diagonal), add 1 to the sum of the row or column. 2. if an entire row or its corresponding column has only s then that vertex is isolated 3. the matrix is diagonally symmetric. Page 5 of 22
6 MODULE 2: NETWORKS AND DECISION MATHEMATICS 9.3 Planar, graphs and Euler s formula A planar graph is a special kind of network or graph. The additional properties of planar graphs will allow us to map two dimensional and even three dimensional objects into graphs. Degenerate graph A graph with no edges is called a degenerate graph (or null graph). Complete graph A graph were all vertices are directly connected to all other vertices without parallel edges or loops is called a complete graph. The figure on the left is a degenerate; the one on the right is complete. How many edges would there be in a complete graph of 6 vertices? Planar graphs Consider the following graphs. Figure a is a planar graph because none of the paths {A, B, C, D, E, F} cross each other. Figure b is apparently not a planar graph because the path (A, D) crosses the path (B, C). Is figure c a planar graph? Page 6 of 22
7 The regions of a planar graph UNDIRECTED GRAPHS AND NETWORKS The above graph can be divided into three regions (faces): region I, region II and region III. Region III is infinite as it continues beyond the bounds of the diagram. Region I and region II have a finite area. Converting a non planar graph Although it may appear that a graph is not planar, by modifying the graph it may become clearly planar. Two method that you could use are: 1. Move the edges 2. Move the vertices Worked Example 6 Convert the graph below to a planar graph. Indicate the faces of the planar graph. The degree of each face is the number of edges defining that region. Worked Example 7 Find the degree of each face of the graph shown in the figure. Page 7 of 22
8 MODULE 2: NETWORKS AND DECISION MATHEMATICS Converting three dimensional solids to planar graphs Worked Example 8 The figure shows a cube with vertices, V = {A, B, C, D, E, F, G, H}. Convert this to a planar graph. Euler s formula The relationship that links vertices, edges and faces of a planar graph is know as Euler s formula. Worked Example 9 Verify Euler s formula for the cube of the last figure in Worked example 8. Note that the cube is a form of prism and all prisms can be converted to planar graphs using the technique of one face collapsing into another. Page 8 of 22
9 UNDIRECTED GRAPHS AND NETWORKS 9.4 Walks, trails, paths, cycles and circuits Walks A walk is a sequence of edges that takes you along some or all of the edges or through some or all of the vertices. There are different ways of naming a walk. For example, consider the figure shown on the right, travelling from node 1 to node 3. A walk could be specified via node 2, namely A B or by specifying the vertices Alternatively, you could the walk C E D, or C F. Each of these routes is a walk. Connected graphs The figure on the left, there is no walk between vertices 1 and 2, nor between vertices 3 and 4, so it is not a connected graph. However, if we add a single edge, as in the figure on the right, between vertices 1 and 2, the entire graph becomes connected. Euler trails A trail is a walk in which no edges are repeated. 1. For an Euler trail to exist, all vertices must be of an even degree or there must be exactly two vertices of odd degree. 2. If the degrees of all the vertices are even numbers, start with any vertex. In this case the starting vertex and ending vertex are the same. 3. If there are two vertices whose degree is an odd number used either as a starting point. The other vertex of odd degree must be the ending point. Worked Example 10 Using the figure, identify an Euler trail. Page 9 of 22
10 MODULE 2: NETWORKS AND DECISION MATHEMATICS Euler circuits In an Euler circuit all the vertices must be of an even degree. If there is an odd degree vertex, an Euler circuit is not possible. Worked Example 11 Find an Euler circuit for the planar graph shown. An Euler circuit algorithm For some networks it may be difficult to determine an Euler circuit. Here is an algorithm (a process) that guarantees an Euler circuit. Consider a network where all vertices ar3e of even degree. Let V = {1, 2, 3, } be the list of vertices. Step 1. Choose a starting vertex from the list V. Call this vertex A. Step 2. Step 3. From vertex A, find the smallest possible path which returns to vertex A. This is a sub circuit of the original network. Let S be the list of vertices in this sub circuit. For each vertex in S, choose a single vertex in turn as the starting vertex of a different sub circuit. It should be as small as possible, and not use any previously used edge. Step 4. For each of these new sub circuits (if there are any), add any new vertices to the list in S. Step 5. Repeat steps 3 and 4 until there are no more new vertices, edges or sub circuits left; that is, the list S and V are the same. Step 6. Join the sub circuits at their intersection points. Page 10 of 22
11 Worked Example 12 Find one possible Euler circuit for the network shown using the Euler circuit algorithm. UNDIRECTED GRAPHS AND NETWORKS Step Instruction Example 1 Choose a starting vertex, and find its smallest sub circuit. 2 Create the list S from the first sub circuit. Find new sub circuit, not using any edges already used. (step 3) 3 Add to the list S. (step 4) 4 Find the new sub circuit. (step 3) 5 Add to list S. Check that all vertices are in the list. (step 4) 6 Form the Euler trail, starting with the first sub circuit, and proceeding through all the other sub circuit at their intersection. (step 6) 7 List the Euler circuit. Path and cycles In Euler trails and circuits each edge was used exactly once, while vertices could be re used. Now, consider the case where it is desirable to use each vertex exactly once. Please note: Not all edges need to be used. There can only be up to 2 vertices with degree 1 (dead ends). These would be the start and/or the finishing vertices. Page 11 of 22
12 MODULE 2: NETWORKS AND DECISION MATHEMATICS Worked Example 13 Determine a Hamiltonian path in the planar graph shown. Hamiltonian cycles When determining a Hamiltonian path, sometimes it is desirable to start and finish with the same vertex. For example, a travelling salesperson may live in one of the towns (vertices) she visits and would like to start and finish at her home town after visiting all the other towns once. This is similar to the concept of an Euler circuit. Worked Example 14 Determine a Hamiltonian cycle in the planar graph shown. 9.5 Trees and their application Graphs and subgraphs A network can also be called a graph. In practice, a graph should have at least 2 vertices and 1 edge. All or part of this graph can be considered as a subgraph. For example, in the figure, the entire network can be considered as a graph, while the path in pink can be considered a subgraph. Another subgraph could be defined by the path A minimum subgraph could be defined by the path 1 2. Often the edges can be assigned some quantity such as distance or cost. If a graph contains quantities, it is then called a weighted graph. Page 12 of 22
13 Trees UNDIRECTED GRAPHS AND NETWORKS Worked Example 15 Determine whether each of the figures is a tree, and if not, explain why not. a. b. c. d. The advantage of trees within a network is that the tree could determine an efficient connection between vertices in the sense that there is a minimum distance, cost or time. Shortest paths Sometimes it may be useful to determine the shortest path between 2 selected vertices of a graph. Worked Example 16 Determine the shortest path between nodes A and F in the figure shown. Possible trees between A F Distance travelled A shortest path algorithm Sometimes it can be difficult to list all the paths between the starting and ending vertex. Here is an algorithm that guarantees the shortest path. Step 1. From the starting vertex, find the shortest path to all other directly connected vertices. Include all such vertices, including the starting one in the list S = {A, B, } Step 2. Choose a vertex directly connected to those in S and find the shortest path to the staring vertex. Step 3. Add the new vertex, V, to the list S. Step 4. Repeat steps 2 and 3 until all vertices are in S. Find the shortest path to the vertex you want. Page 13 of 22
14 MODULE 2: NETWORKS AND DECISION MATHEMATICS Worked Example 17 Find the shortest path from vertex 1 to vertex From 1 find shortest path to each vertices directly connected to it From To Via Distance Shortest path to 2. Determine set of vertices in S 3. Choose a vertex directly connected to one in S Select vertex 5 4. Add new vertex to the S list 5. Choose a vertex directly connected to one in S Select vertex 6 6. Add new vertex to the S list 7. Choose a vertex directly connected to one in S Select vertex 7 8. Add new vertex to the S list 9. Choose a vertex directly connected to one in S Select vertex Add new vertex to the S list 11. Choose last vertex 9 Select vertex Add new vertex to the S list 13. Stop once all vertices are listed Page 14 of 22
15 UNDIRECTED GRAPHS AND NETWORKS Spanning trees Spanning trees include all vertices. Consider the following network. If the vertices are a location/town /house etc. and the distances are shown on the edges. Then the figures below show the possible trees. Note that each of these trees included all the vertices of the original network. These trees are called spanning trees because of this property. Page 15 of 22
16 MODULE 2: NETWORKS AND DECISION MATHEMATICS Minimum spanning tree and Prim s algorithm Minimum spanning trees connect all vertices with the smallest distance. Prim s algorithm Step 1. Step 2. Step 3. Step 4. Worked Example 18 Choose the edge with the minimum value. If 2 or more edges are the smallest, choose any of these. Inspect the next edges connected to the two vertices on the ends of your original edge. Choose the smallest edge. Inspect all included vertices so far and choose the smallest edge leading from any vertex. Repeat until all vertices are included. Determine the minimum spanning tree for the network representing footpaths in a school campus. Page 16 of 22
17 Maximum spanning tree Worked Example 19 UNDIRECTED GRAPHS AND NETWORKS Maximum spanning trees connect all vertices with the largest distance. The figure shown represents a telephone network connecting 6 towns, A, B,, F. The numbered edges represent the capacity of the telephone connection between the towns connected, that is, the maximum number of calls that can be made at the same time along that edge. A telephone engineer wishes to determine the maximum capacity of the system in terms of a tree connecting all the towns so that calls can be routed along that tree. Page 17 of 22
18 MODULE 2: NETWORKS AND DECISION MATHEMATICS Dijkstra s algorithm Another method for determining the shortest path between a given vertex and each of the other vertices is Dijkstra s algorithm. Dijkstra s algorithm Example 1 The weights in the graph opposite show the distances, in kilometres, by road between the towns. The algorithm will be used to find the shortest path between St Andrews (S ) and Toolangi (T ). Step 1: Create a table 1. Write the starting vertex as the first row vertex. 2. Write the other vertices in the network as column vertices (the order is unimportant). S K Y T H Step 2: Complete the first row Look at the graph to find the distance from the starting vertex (the row vertex in this example S) to the vertices that are directly connected to it (a column vertex). 1. Write down the distance from the row vertex to a column vertex, directly underneath the column vertex as shown. 2. If a vertex is not directly connected to the starting vertex, mark a cross (x). 3. Look for the smallest number in the first row and draw a box around it. If there are two or more the same, any one can be chosen. 4. The column vertex for this boxed number becomes the next row vertex. K Y T H S X X K Y T H S X X K Step 3: Complete the further rows 1. Copy all boxed numbers into the next row. K Y T H S X X K 13 Page 18 of 22
19 2. For the remaining columns, add the boxed numbers for the row vertex to the distance from the row vertex to the column vertex. a) If the value is greater than the value above it, ignore the new value and copy the existing one. b) If the value is less than or equal to the value above it in the column, write down new value. c) If the row vertex is not directly connected to the column vertex, mark with a cross (x). 3. Look for the smallest unboxed number in the row and draw a box around it. 4. The column vertex for this new boxed number becomes the next row vertex. 5. Repeat step 3 until the destination vertex value has a boxed around it. UNDIRECTED GRAPHS AND NETWORKS K Y T H S X X K X K Y T H S X X K X Y K Y T H S X X K X Y Step 4: Backtrack to identify the shortest path and its length 1. Start at the box value for the destination vertex. This is the length of the shortest path from the starting vertex to the destination vertex. 2. Draw a line up the column to the last number that is the same as the box number (it does not have to have a box around it). 3. Look at the row vertex for this number and draw a horizontal line to the column for this vertex. 4. Repeat until the starting vertex is reached. 5. The horizontal lines in the table indicate the shortest path. K Y T H S X X K X Y Page 19 of 22
20 MODULE 2: NETWORKS AND DECISION MATHEMATICS Dijkstra s algorithm Example 2 Find the shortest path from A to F in the weighted graph shown using Dijkstra s algorithm Step 1: Create a table 1. Write the starting vertex as the first row vertex. 2. Write the other vertices in the network as column vertices (the order is unimportant). Step 2: Complete the first row Look at the graph to find the distance from the starting vertex (the row vertex in this example S) to the vertices that are directly connected to it (a column vertex). 1. Write down the distance from the row vertex to a column vertex, directly underneath the column vertex as shown. 2. If a vertex is not directly connected to the starting vertex, mark a cross (x). 3. Look for the smallest number in the first row and draw a box around it. If there are two or more the same, any one can be chosen. 4. The column vertex for this boxed number becomes the next row vertex. Step 3: Complete the further rows 1. Copy all boxed numbers into the next row. 2. For the remaining columns, add the boxed numbers for the row vertex to the distance from the row vertex to the column vertex. a) If the value is greater than the value above it, ignore the new value and copy the existing one. b) If the value is less than or equal to the value above it in the column, write down new value. c) If the row vertex is not directly connected to the column vertex, mark with a cross (x). 3. Look for the smallest unboxed number in the row and draw a box around it. 4. The column vertex for this new boxed number becomes the next row vertex. 5. Repeat step 3 until the destination vertex value has a boxed around it. A A A A A A B C D E F B C D E F B C D E F B C D E F B C D E F B C D E F Page 20 of 22
21 Step 4: Backtrack to identify the shortest path and its length 1. Start at the box value for the destination vertex. This is the length of the shortest path from the starting vertex to the destination vertex. 2. Draw a line up the column to the last number that is the same as the box number (it does not have to have a box around it). 3. Look at the row vertex for this number and draw a horizontal line to the column for this vertex. 4. Repeat until the starting vertex is reached. 5. The horizontal lines in the table indicate the shortest path. A UNDIRECTED GRAPHS AND NETWORKS B C D E F Worked Example 20 (STUDENT PRACTICE) Determine the shortest path from A to E, where the distance are in kilometres, by using Dijkstra s algorithm in tabular form. Page 21 of 22
22 MODULE 2: NETWORKS AND DECISION MATHEMATICS Additional Dijkstra s Algorithm Questions Page 22 of 22
Further Mathematics 2016 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks
Further Mathematics 2016 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks Key knowledge the conventions, terminology, properties and types of graphs; edge, face, loop,
More informationUndirected Network Summary
Undirected Network Summary Notice that the network above has multiple edges joining nodes a to d and the network has a loop at node d. Also c is called an isolated node as it is not connected to any other
More informationTopic 10 Part 2 [474 marks]
Topic Part 2 [474 marks] The complete graph H has the following cost adjacency matrix Consider the travelling salesman problem for H a By first finding a minimum spanning tree on the subgraph of H formed
More informationGRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS
GRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS DR. ANDREW SCHWARTZ, PH.D. 10.1 Graphs and Graph Models (1) A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes)
More informationMEI Further Mathematics Support Programme
Further Mathematics Support Programme the Further Mathematics Support Programme www.furthermaths.org.uk Modelling and problem solving with Networks Sharon Tripconey Let Maths take you Further Nov 2009
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. An Introduction to Graph Theory
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics An Introduction to Graph Theory. Introduction. Definitions.. Vertices and Edges... The Handshaking Lemma.. Connected Graphs... CutPoints and Bridges.
More informationWUCT121. Discrete Mathematics. Graphs
WUCT121 Discrete Mathematics Graphs WUCT121 Graphs 1 Section 1. Graphs 1.1. Introduction Graphs are used in many fields that require analysis of routes between locations. These areas include communications,
More informationWorksheet for the Final Exam  Part I. Graphs
Worksheet for the Final Exam  Part I. Graphs Date and Time: May 10 2012 Thursday 11:50AM~1:50PM Location: Eng 120 Start with the SelfTest Exercises (pp.816) in Prichard. 1. Give the adjacency matrix
More informationChapter 14. Graphs Pearson AddisonWesley. All rights reserved 14 A1
Chapter 14 Graphs 2011 Pearson AddisonWesley. All rights reserved 14 A1 Terminology G = {V, E} A graph G consists of two sets A set V of vertices, or nodes A set E of edges A subgraph Consists of a subset
More informationAn Introduction to Graph Theory
An Introduction to Graph Theory CIS0082 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 12:00, Friday 17 th February 2012 Outline 1 Graphs 2 Paths and cycles 3 Graphs and
More informationGraphs Definitions. Gunnar Gotshalks. GraphDefinitions 1
Graphs Definitions GraphDefinitions 1 Examples of Graphs Street maps» Vertices are the intersections» Edges are the streets Power line network» Vertices are the houses & power stations» Edges are the power
More information1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G))
10 Graphs 10.1 Graphs and Graph Models 1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G)) 2. an edge is present, say e = {u,
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 informationChapter 5: Euler Paths and Circuits The Mathematics of Getting Around
1 Finite Math A Chapter 5: Euler Paths and Circuits The Mathematics of Getting Around Academic Standards Covered in this Chapter: *************************************************************************************
More informationGraph Theory. Part of Texas Counties.
Graph Theory Part of Texas Counties. We would like to visit each of the above counties, crossing each county only once, starting from Harris county. Is this possible? This problem can be modeled as a graph.
More informationMEI Conference 2009: D2
D Networks MEI Conference 009: D Travelling Salesperson Problem 7 A B Route Inspection Problems (Chinese Postman) () A 7 B () 4 () C Identify the odd vertices in the network Consider all the routes joining
More informationAssignment 4 Solutions of graph problems
Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of
More informationDS UNIT 4. Matoshri College of Engineering and Research Center Nasik Department of Computer Engineering Discrete Structutre UNIT  IV
Sr.No. Question Option A Option B Option C Option D 1 2 3 4 5 6 Class : S.E.Comp Which one of the following is the example of non linear data structure Let A be an adjacency matrix of a graph G. The ij
More informationChapter 5: Euler Paths and Circuits The Mathematics of Getting Around
1 Finite Math A Chapter 5: Euler Paths and Circuits The Mathematics of Getting Around Academic Standards Covered in this Chapter: *************************************************************************************
More informationVarying Applications (examples)
Graph Theory Varying Applications (examples) Computer networks Distinguish between two chemical compounds with the same molecular formula but different structures Solve shortest path problems between cities
More informationGraph (1A) Young Won Lim 4/19/18
Graph (1A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationDiscrete mathematics II.  Graphs
Emil Vatai April 25, 2018 Basic definitions Definition of an undirected graph Definition (Undirected graph) An undirected graph or (just) a graph is a triplet G = (ϕ, E, V ), where V is the set of vertices,
More informationChapter 1 Graph Theory
Chapter Graph Theory  Representations of Graphs Graph, G=(V,E): It consists of the set V of vertices and the set E of edges. If each edge has its direction, the graph is called the directed graph (digraph).
More informationINTRODUCTION TO GRAPH THEORY. 1. Definitions
INTRODUCTION TO GRAPH THEORY D. JAKOBSON 1. Definitions A graph G consists of vertices {v 1, v 2,..., v n } and edges {e 1, e 2,..., e m } connecting pairs of vertices. An edge e = (uv) is incident with
More informationMath Summer 2012
Math 481  Summer 2012 Final Exam You have one hour and fifty minutes to complete this exam. You are not allowed to use any electronic device. Be sure to give reasonable justification to all your answers.
More informationElements of Graph Theory
Elements of Graph Theory Quick review of Chapters 9.1 9.5, 9.7 (studied in Mt1348/2008) = all basic concepts must be known New topics we will mostly skip shortest paths (Chapter 9.6), as that was covered
More informationMathematics and Statistics, Part A: Graph Theory Problem Sheet 1, lectures 14
1. Draw Mathematics and Statistics, Part A: Graph Theory Problem Sheet 1, lectures 14 (i) a simple graph. A simple graph has a nonempty vertex set and no duplicated edges. For example sketch G with V
More informationCS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS
CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1. Define Graph. A graph G = (V, E) consists
More informationCS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK
CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK 1. Define Graph. 2. Define Simple graph. 3. Write few problems
More information0.0.1 Network Analysis
Graph Theory 0.0.1 Network Analysis Prototype Example: In Algonquian Park the rangers have set up snowmobile trails with various stops along the way. The system of trails is our Network. The main entrance
More informationGraph and Digraph Glossary
1 of 15 31.1.2004 14:45 Graph and Digraph Glossary A B C D E F G H IJ K L M N O PQ R S T U V WZ Acyclic Graph A graph is acyclic if it contains no cycles. Adjacency Matrix A 01 square matrix whose
More informationIntroduction to Graph Theory
Introduction to Graph Theory Tandy Warnow January 20, 2017 Graphs Tandy Warnow Graphs A graph G = (V, E) is an object that contains a vertex set V and an edge set E. We also write V (G) to denote the vertex
More informationAlgorithms. Graphs. Algorithms
Algorithms Graphs Algorithms Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Algorithms 1 Graphs Vertices: Nodes, points, computers, users, items,...
More informationSimple graph Complete graph K 7. Non connected graph
A graph G consists of a pair (V; E), where V is the set of vertices and E the set of edges. We write V (G) for the vertices of G and E(G) for the edges of G. If no two edges have the same endpoints we
More informationFoundations of Discrete Mathematics
Foundations of Discrete Mathematics Chapter 12 By Dr. Dalia M. Gil, Ph.D. Trees Tree are useful in computer science, where they are employed in a wide range of algorithms. They are used to construct efficient
More information4. (a) Draw the Petersen graph. (b) Use Kuratowski s teorem to prove that the Petersen graph is nonplanar.
UPPSALA UNIVERSITET Matematiska institutionen Anders Johansson Graph Theory Frist, KandMa, IT 010 10 1 Problem sheet 4 Exam questions Solve a subset of, say, four questions to the problem session on friday.
More informationQuestion. Why is the third shape not convex?
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationChapter 9 Graph Algorithms
Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many reallife problems can be if not careful with data structures 3 Definitions an undirected graph G = (V, E) is a
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 informationChapter 9 Graph Algorithms
Introduction graph theory useful in practice represent many reallife problems can be if not careful with data structures Chapter 9 Graph s 2 Definitions Definitions an undirected graph is a finite set
More informationAssignments are handed in on Tuesdays in even weeks. Deadlines are:
Tutorials at 2 3, 3 4 and 4 5 in M413b, on Tuesdays, in odd weeks. i.e. on the following dates. Tuesday the 28th January, 11th February, 25th February, 11th March, 25th March, 6th May. Assignments are
More informationGraph Theory. 1 Introduction to Graphs. Martin Stynes Department of Mathematics, UCC January 26, 2011
Graph Theory Martin Stynes Department of Mathematics, UCC email: m.stynes@ucc.ie January 26, 2011 1 Introduction to Graphs 1 A graph G = (V, E) is a nonempty set of nodes or vertices V and a (possibly
More informationAlgorithms: Graphs. Amotz BarNoy. Spring 2012 CUNY. Amotz BarNoy (CUNY) Graphs Spring / 95
Algorithms: Graphs Amotz BarNoy CUNY Spring 2012 Amotz BarNoy (CUNY) Graphs Spring 2012 1 / 95 Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Amotz
More information1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D.
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationIntroduction III. Graphs. Motivations I. Introduction IV
Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg
More informationInstant Insanity Instructor s Guide Makeit and Takeit Kit for AMTNYS 2006
Instant Insanity Instructor s Guide Makeit and Takeit Kit for AMTNYS 2006 THE KIT: This kit contains materials for two Instant Insanity games, a student activity sheet with answer key and this instructor
More informationNotebook Assignments
Notebook Assignments These six assignments are a notebook using techniques from class in the single concrete context of graph theory. This is supplemental to your usual assignments, and is designed for
More informationTWO CONTRIBUTIONS OF EULER
TWO CONTRIBUTIONS OF EULER SIEMION FAJTLOWICZ. MATH 4315 Eulerian Tours. Although some mathematical problems which now can be thought of as graphtheoretical, go back to the times of Euclid, the invention
More informationThe University of Sydney MATH2969/2069. Graph Theory Tutorial 2 (Week 9) 2008
The University of Sydney MATH99/09 Graph Theory Tutorial (Week 9) 00. Show that the graph on the left is Hamiltonian, but that the other two are not. To show that the graph is Hamiltonian, simply find
More informationExamples of Tasks from Course 1, Unit 4
Examples of Tasks from Course 1, Unit 4 What Solutions are Available? Lesson 1: page 258, Modeling Task 1; page 260, Modeling Task 4; page 261, Organizing Task 1; page 271, Modeling Task 2; page 272, Modeling
More informationv V Question: How many edges are there in a graph with 10 vertices each of degree 6?
ECS20 Handout Graphs and Trees March 4, 2015 (updated 3/9) Notion of a graph 1. A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of pairs of elements of V called edges.
More information2 hours THE UNIVERSITY OF MANCHESTER. 22 May :00 16:00
2 hours THE UNIVERSITY OF MANCHESTER DISCRETE MATHEMATICS 22 May 2015 14:00 16:00 Answer ALL THREE questions in Section A (30 marks in total) and TWO of the THREE questions in Section B (50 marks in total).
More informationHonors ICM Graph Theory Unit 7 Homework Packet Homework Day 1
Honors ICM Graph Theory Unit 7 Homework Packet Homework Day 1 Name Period: 6. Construct a graph with three critical paths. 7. Determine the minimum project time and the critical path for the following
More informationEECS 203 Lecture 20. More Graphs
EECS 203 Lecture 20 More Graphs Admin stuffs Last homework due today Office hour changes starting Friday (also in Piazza) Friday 6/17: 25 Mark in his office. Sunday 6/19: 25 Jasmine in the UGLI. Monday
More informationGraph Theory CS/Math231 Discrete Mathematics Spring2015
1 Graphs Definition 1 A directed graph (or digraph) G is a pair (V, E), where V is a finite set and E is a binary relation on V. The set V is called the vertex set of G, and its elements are called vertices
More informationChapter 9 Graph Algorithms
Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many reallife problems can be slow if not careful with data structures 3 Definitions an undirected graph G = (V, E)
More informationChapter 2 Graphs. 2.1 Definition of Graphs
Chapter 2 Graphs Abstract Graphs are discrete structures that consist of vertices and edges connecting some of these vertices. Graphs have many applications in Mathematics, Computer Science, Engineering,
More information11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions
Introduction Chapter 9 Graph Algorithms graph theory useful in practice represent many reallife problems can be slow if not careful with data structures 2 Definitions an undirected graph G = (V, E) is
More informationMajority and Friendship Paradoxes
Majority and Friendship Paradoxes Majority Paradox Example: Small town is considering a bond initiative in an upcoming election. Some residents are in favor, some are against. Consider a poll asking the
More informationMAS 341: GRAPH THEORY 2016 EXAM SOLUTIONS
MS 41: PH THEOY 2016 EXM SOLUTIONS 1. Question 1 1.1. Explain why any alkane C n H 2n+2 is a tree. How many isomers does C 6 H 14 have? Draw the structure of the carbon atoms in each isomer. marks; marks
More informationGraph Overview (1A) Young Won Lim 5/9/18
Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationAbout the Tutorial. Audience. Prerequisites. Disclaimer & Copyright. Graph Theory
About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a readerfriendly style, it covers the types of graphs, their properties, trees, graph traversability,
More informationOutline. Introduction. Representations of Graphs Graph Traversals. Applications. Definitions and Basic Terminologies
Graph Chapter 9 Outline Introduction Definitions and Basic Terminologies Representations of Graphs Graph Traversals Breadth first traversal Depth first traversal Applications Single source shortest path
More informationOrdinary Differential Equation (ODE)
Ordinary Differential Equation (ODE) INTRODUCTION: Ordinary Differential Equations play an important role in different branches of science and technology In the practical field of application problems
More informationMC 302 GRAPH THEORY 10/1/13 Solutions to HW #2 50 points + 6 XC points
MC 0 GRAPH THEORY 0// Solutions to HW # 0 points + XC points ) [CH] p.,..7. This problem introduces an important class of graphs called the hypercubes or kcubes, Q, Q, Q, etc. I suggest that before you
More informationChapter 3: Paths and Cycles
Chapter 3: Paths and Cycles 5 Connectivity 1. Definitions: Walk: finite sequence of edges in which any two consecutive edges are adjacent or identical. (Initial vertex, Final vertex, length) Trail: walk
More information08. First and second degree equations
08. First and second degree equations GRAPH THEORY Based on Chris K. Caldwell work: http://primes.utm.edu/cgibin/caldwell/tutor/graph/index.html INTRODUCTION Consider the next problem: Old Königsberg
More informationGraph Theory II. PoShen Loh. June edges each. Solution: Spread the n vertices around a circle. Take parallel classes.
Graph Theory II PoShen Loh June 009 1 Warmup 1. Let n be odd. Partition the edge set of K n into n matchings with n 1 edges each. Solution: Spread the n vertices around a circle. Take parallel classes..
More informationBIL694Lecture 1: Introduction to Graphs
BIL694Lecture 1: Introduction to Graphs Lecturer: Lale Özkahya Resources for the presentation: http://www.math.ucsd.edu/ gptesler/184a/calendar.html http://www.inf.ed.ac.uk/teaching/courses/dmmr/ Outline
More informationCombinatorics Summary Sheet for Exam 1 Material 2019
Combinatorics Summary Sheet for Exam 1 Material 2019 1 Graphs Graph An ordered threetuple (V, E, F ) where V is a set representing the vertices, E is a set representing the edges, and F is a function
More information1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT
DECISION 1 Revision Notes 1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT Make sure you show comparisons clearly and label each pass First Pass 8 4 3 6 1 4 8 3 6 1 4 3 8 6 1 4 3 6 8 1
More informationBasic Combinatorics. Math 40210, Section 01 Fall Homework 4 Solutions
Basic Combinatorics Math 40210, Section 01 Fall 2012 Homework 4 Solutions 1.4.2 2: One possible implementation: Start with abcgfjiea From edge cd build, using previously unmarked edges: cdhlponminjkghc
More informationPlanar Graph (7A) Young Won Lim 5/21/18
Planar Graph (7A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationStudent Name and ID Number. MATH 3012 Final Exam, December 11, 2014, WTT
MATH 3012 Final Exam, December 11, 2014, WTT Student Name and ID Number 1. Consider the 11element set X consisting of the three capital letters {A, B, C} and the eight digits {0, 1, 2,..., 7}. a. How
More informationHow can we lay cable at minimum cost to make every telephone reachable from every other? What is the fastest route between two given cities?
1 Introduction Graph theory is one of the most indemand (i.e. profitable) and heavilystudied areas of applied mathematics and theoretical computer science. May graph theory questions are applied in this
More informationPATH FINDING AND GRAPH TRAVERSAL
GRAPH TRAVERSAL PATH FINDING AND GRAPH TRAVERSAL Path finding refers to determining the shortest path between two vertices in a graph. We discussed the Floyd Warshall algorithm previously, but you may
More informationLecture 10. Elementary Graph Algorithm Minimum Spanning Trees
Lecture 10. Elementary Graph Algorithm Minimum Spanning Trees T. H. Cormen, C. E. Leiserson and R. L. Rivest Introduction to Algorithms, 3rd Edition, MIT Press, 2009 Sungkyunkwan University Hyunseung Choo
More informationMT365 Examination 2017 Part 1 Solutions Part 1
MT xamination 0 Part Solutions Part Q. G (a) Number of vertices in G =. Number of edges in G = (i) The graph G is simple no loops or multiple edges (ii) The graph G is not regular it has vertices of deg.,
More informationLecture 5: Graphs. Rajat Mittal. IIT Kanpur
Lecture : Graphs Rajat Mittal IIT Kanpur Combinatorial graphs provide a natural way to model connections between different objects. They are very useful in depicting communication networks, social networks
More informationGraphs. The ultimate data structure. graphs 1
Graphs The ultimate data structure graphs 1 Definition of graph Nonlinear data structure consisting of nodes & links between them (like trees in this sense) Unlike trees, graph nodes may be completely
More informationPlanar Graph (7A) Young Won Lim 6/20/18
Planar Graph (7A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
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 informationDegree of nonsimple graphs. Chemistry questions. Degree Sequences. Pigeon party.
1. WEEK 1 PROBLEMS 1.1. Degree of nonsimple graphs. In the lecture notes we defined the degree d(v) of a vertex v to be the number of vertices adjacent to v. To see why Euler s theorem doesn t hold for
More informationInstructor: Paul Zeitz, University of San Francisco
Berkeley Math Circle Graph Theory and Ramsey Theory Instructor: Paul Zeitz, University of San Francisco (zeitz@usfca.edu) Definitions 1 A graph is a pair (V,E), where V is a finite set and E is a set of
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION MH 1301 DISCRETE MATHEMATICS TIME ALLOWED: 2 HOURS
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION 20152016 MH 1301 DISCRETE MATHEMATICS May 2016 TIME ALLOWED: 2 HOURS INSTRUCTIONS TO CANDIDATES 1. This examination paper contains FIVE (5) questions
More informationProblem 1. Which of the following is true of functions =100 +log and = + log? Problem 2. Which of the following is true of functions = 2 and =3?
Multiplechoice Problems: Problem 1. Which of the following is true of functions =100+log and =+log? a) = b) =Ω c) =Θ d) All of the above e) None of the above Problem 2. Which of the following is true
More informationCHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism
CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition 1.1.1. A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented
More informationGraphs (MTAT , 6 EAP) Lectures: Mon 1416, hall 404 Exercises: Wed 1416, hall 402
Graphs (MTAT.05.080, 6 EAP) Lectures: Mon 1416, hall 404 Exercises: Wed 1416, hall 402 homepage: http://courses.cs.ut.ee/2012/graafid (contains slides) For grade: Homework + three tests (during or after
More informationCHAPTER 10 GRAPHS AND TREES. Alessandro Artale UniBZ  artale/z
CHAPTER 10 GRAPHS AND TREES Alessandro Artale UniBZ  http://www.inf.unibz.it/ artale/z SECTION 10.1 Graphs: Definitions and Basic Properties Copyright Cengage Learning. All rights reserved. Graphs: Definitions
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 informationAlgorithm Analysis Graph algorithm. ChungAng University, Jaesung Lee
Algorithm Analysis Graph algorithm ChungAng University, Jaesung Lee Basic definitions Graph = (, ) where is a set of vertices and is a set of edges Directed graph = where consists of ordered pairs
More informationGrade 7/8 Math Circles Graph Theory  Solutions October 13/14, 2015
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Graph Theory  Solutions October 13/14, 2015 The Seven Bridges of Königsberg In
More information6.2. Paths and Cycles
6.2. PATHS AND CYCLES 85 6.2. Paths and Cycles 6.2.1. Paths. A path from v 0 to v n of length n is a sequence of n+1 vertices (v k ) and n edges (e k ) of the form v 0, e 1, v 1, e 2, v 2,..., e n, v n,
More informationSections 5.2, 5.3. & 5.4
MATH 11008: Graph Theory Terminology Sections 5.2, 5.3. & 5.4 Routing problem: A routing problem is concerned with finding ways to route the delivery of good and/or services to an assortment of destinations.
More informationMath 311. Trees Name: A Candel CSUN Math
1. A simple path in a graph is a path with no repeated edges. A simple circuit is a circuit without repeated edges. 2. Trees are special kinds of graphs. A tree is a connected graph with no simple circuits.
More informationCharacterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)
S72.2420/T79.5203 Basic Concepts 1 S72.2420/T79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G
More informationSimple Graph. General Graph
Graph Theory A graph is a collection of points (also called vertices) and lines (also called edges), with each edge ending at a vertex In general, it is allowed for more than one edge to have the same
More informationHomework 5: Graphs, Minimum Spanning Trees, and Dijkstra ShortestPath
Homework 5: Graphs, Minimum Spanning Trees, and Dijkstra ShortestPath 1. (4 points) A graph is Hamiltonian if there is a cycle in the graph visiting each vertex exactly once. Give an example of an Eulerian
More informationFundamental Properties of Graphs
Chapter three In many reallife 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 informationMathematics MD01. General Certificate of Education Advanced Subsidiary Examination. Unit Decision 1
Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Decision 1 General Certificate of Education Advanced Subsidiary Examination
More information