Discrete Structures CISC 2315 FALL Graphs & Trees

Size: px
Start display at page:

Download "Discrete Structures CISC 2315 FALL Graphs & Trees"

Transcription

1 Discrete Structures CISC 2315 FALL 2010 Graphs & Trees

2 Graphs A graph is a discrete structure, with discrete components

3 Components of a Graph edge vertex (node)

4 Vertices A graph G = (V, E), where V is the set of all the vertices in the graph, and E is the set of all edges in the graph. The elements of V are typically named u or v.

5 Types of Edges Undirected edges: In this case, there is a function f from E to {{ u, v} u, v V} Note unordered An edge is a loop if f(e) = {u, u} = {u} for some u in V. loop Undirected graph

6 Types of Edges Directed edges: Note ordered In this case, there is a function f from E to {( u, v) u, v V} An edge is a loop if f(e) = (u, u) for some u in V. loop Directed graph

7 Graph Terminology Two vertices u and v in an undirected graph G are adjacent (neighbors) in G if {u,v} is an edge of G. If e={u,v}, then the edge e is called incident with the vertices u and v. Edge e is said to connect u and v. Also, vertices u and v are called endpoints of the edge {u,v}.

8 Graph Terminology u e v Vertices u and v are adjacent. Edge e is incident with u and v; e connects u and v. Vertices u and v are endpoints of e.

9 Degree The degree of a vertex in an undirected graph is the number of edges incident with it. A loop contributes twice to the degree. A vertex of degree zero is called isolated. A vertex of degree 1 is called pendant.

10 Graph Terminology u e v What is the deg(u)? What is the deg(v)?

11 Graph Terminology In a directed graph G with edge (u,v), u is said to be adjacent to v and v is said to be adjacent from u. Vertex u is the initial vertex and v is the terminal (end) vertex of (u,v). The initial and terminal vertices of a loop are the same.

12 Graph Terminology u e v Vertex u is adjacent to v and v is adjacent from u. Vertex u is the initial vertex and v is the terminal vertex of edge (u,v).

13 Degree The in-degree of a vertex v, deg - (v) in a directed graph is the number of edges with v as their terminal vertex. The out-degree of v, deg + (v), is the number of edges with v as their initial vertex. A loop contributes 1 to both the in-degree and out-degree.

14 Graph Terminology u e v What is the in-degree of u? the out-degree of u? What is the in-degree of v? the out-degree of v?

15 Simple Graphs A graph is simple if it has only one edge connecting each pair of vertices.

16 Simple Graphs simple not simple

17 Bipartite Graph A simple graph G=(V,E) is bipartite if V can be partitioned into disjoint sets V 1 and V 2 such that every edge in the graph connects a vertex in V 1 and a vertex in V 2. No edge in G connects either two vertices in V 1 or two vertices in V 2.

18 Bipartite Graph

19 Union of Graphs The union of two simple graphs G 1 =(V 1,E 1 ) and G 2 =(V 2,E 2 ) is the simple graph G G V V, E ) 1 2 ( E2

20 Union of a Graphs

21 Matrices Used to represent graphs in a computer program. A matrix is a rectangular array of numbers. a 11 a 12 a 1n A = a 21 a 22 a 2n... a 1m a 2m a mn This is an m x n matrix.

22 Adjacent Vertices in a Graph a c e g vertex a b c b,c,d,g a,d a,d,e adjacent vertices b d d e f? a,b,c,e,f c,d,f,g g? f

23 Adjacency Matrix to Represent a Graph a b c d e f g a c e g a b c d b d e f f g

24 Incidence to Represent a Graph e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 a b e 7 e 2 e 3 e 4 c d e 5 e 6 e 1 e e 9 e 8 e 10 e 11 g a b c d e f f g

25 Path A path is a sequence of edges that begins with a vertex of the graph and travels along edges of the graph, always connecting pairs of adjacent vertices. A path of length n from u to v in an undirected graph is a sequence of edges e 1,,e n that begins with u and ends with v. The path is a circuit if u=v. The path passes through the vertices that are visited, and it traverses the edges on the path. A path or circuit is simple if it doesn t contain the same edge more than once.

26 Path in Matrix an Undirected Graph u v

27 Connectedness An undirected graph is connected if there is a path between every pair of distinct vertices in the graph.

28 Is this graph connected?

29 Trees Note that a tree is a connected undirected graph that has no simple circuits.

30 Path A path of length n from u to v in a directed graph is a sequence of directed edges e 1,,e n that begins with u and ends with v. The path is a circuit if u=v. The path passes through the vertices that are visited, and it traverses the edges on the path. A path or circuit is simple if it doesn t contain the same edge more than once.

31 Circuit in a Directed Graph u=v

32 Connectedness A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph. A directed graph is weakly connected if there is a path between every two vertices in the underlying undirected graph.

33 Is this graph strongly/weakly connected?

34 Example Applications of Graphs Designing airplane routes Modeling the interconnections and information flow in local and wide area computer networks Models of ecologies Finding the shortest path between two locations (uses distances along the edges) Solving search problems in artificial intelligence

35 Lists - revisited A list is a finite ordered sequence of zero or more elements that can be repeated. The difference between lists and tuples is in what parts can be randomly accessed. Head (L) and Tail (L) is h( w, x, y, z ) w, t( w, x, y, z ) x, y, z. Memory representation of a list in the computer head (L) = b tail (L) L b c d e cons (a,l) a

36 Computer representation of a tree T a, b, c, d, e. a a b c d b c d e e

37 Binary Tree T d, b, e, a, c. a a b c b c d e d e Binary trees can be used to represent sets whose elements have some ordering. Such a tree is called a binary search tree and has the property that for each node Of the tree, each element in its left subtree precedes the node element and each Element in its right subtree succeeds the node element.

38 Spanning Trees Kruskal s Algorithm f 1 e g a d 2 b c L= {{a,b},{c,d},{d,g},{e,f},{f,g},{a,f},{b,c},{c,g},{d,e},{e,g},{a,g},{b,g}} Spanning Tree T Equivalance Classes {} {a},{b},{c},{d},{e},{f},{g} {{a,b}} {{a,b},{c,d}} {{a,b},{c,d},{d,g}} {{a,b},{c,d},{d,g},{e,f}} {{a,b},{c,d},{d,g},{e,f},{f,g}} {a,b},{c},{d},{e},{f},{g} {a,b},{c,d},{e},{f},{g} {a,b},{c,d,g},{e},{f} {a,b},{c,d,g},{e,f} {a,b},{c,d,e,f,g} {{a,b},{c,d},{d,g},{e,f},{f,g},{a,f}} {a,b,c,d,e,f,g} Kruskal s Algorithm: 1) Sort the edges of the graph by weight, and let L be the sorted list 2) Let T be the minimal spanning tree and initialize T : = 0. 3) For each vertex v of the graph, create the equivalence class [v] = {v}. 4) while there are 2 or more equivalence classes do Let {a,b} be the edge at the head of L: L := tail (L); if [a} not = [b] then T:= T U {{a,b}}; Replace the equivalence classes [a] and [b] by [a] U [b] fi od

Graphs. Introduction To Graphs: Exercises. Definitions:

Graphs. Introduction To Graphs: Exercises. Definitions: Graphs Eng.Jehad Aldahdooh Introduction To Graphs: Definitions: A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated

More information

Varying Applications (examples)

Varying 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 information

Paths. Path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.

Paths. Path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph. Paths Path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph. Formal Definition of a Path (Undirected) Let n be a nonnegative integer

More information

CPCS Discrete Structures 1

CPCS Discrete Structures 1 Let us switch to a new topic: Graphs CPCS 222 - Discrete Structures 1 Introduction to Graphs Definition: A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs

More information

Introduction to Graphs

Introduction to Graphs Graphs Introduction to Graphs Graph Terminology Directed Graphs Special Graphs Graph Coloring Representing Graphs Connected Graphs Connected Component Reading (Epp s textbook) 10.1-10.3 1 Introduction

More information

CS200: Graphs. Prichard Ch. 14 Rosen Ch. 10. CS200 - Graphs 1

CS200: Graphs. Prichard Ch. 14 Rosen Ch. 10. CS200 - Graphs 1 CS200: Graphs Prichard Ch. 14 Rosen Ch. 10 CS200 - Graphs 1 Graphs A collection of nodes and edges What can this represent? n A computer network n Abstraction of a map n Social network CS200 - Graphs 2

More information

Graphs. Pseudograph: multiple edges and loops allowed

Graphs. Pseudograph: multiple edges and loops allowed Graphs G = (V, E) V - set of vertices, E - set of edges Undirected graphs Simple graph: V - nonempty set of vertices, E - set of unordered pairs of distinct vertices (no multiple edges or loops) Multigraph:

More information

CHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism

CHAPTER 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 information

Introduction III. Graphs. Motivations I. Introduction IV

Introduction 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 information

MTL 776: Graph Algorithms. B S Panda MZ 194

MTL 776: Graph Algorithms. B S Panda MZ 194 MTL 776: Graph Algorithms B S Panda MZ 194 bspanda@maths.iitd.ac.in bspanda1@gmail.com Lectre-1: Plan Definitions Types Terminology Sb-graphs Special types of Graphs Representations Graph Isomorphism Definitions

More information

Elements of Graph Theory

Elements 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 information

GRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS

GRAPHS, 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 information

Discrete Mathematics (2009 Spring) Graphs (Chapter 9, 5 hours)

Discrete Mathematics (2009 Spring) Graphs (Chapter 9, 5 hours) Discrete Mathematics (2009 Spring) Graphs (Chapter 9, 5 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University June 1, 2009 9.1 Graphs and Graph Models What are Graphs? General meaning

More information

DEFINITION OF GRAPH GRAPH THEORY GRAPHS ACCORDING TO THEIR VERTICES AND EDGES EXAMPLE GRAPHS ACCORDING TO THEIR VERTICES AND EDGES

DEFINITION OF GRAPH GRAPH THEORY GRAPHS ACCORDING TO THEIR VERTICES AND EDGES EXAMPLE GRAPHS ACCORDING TO THEIR VERTICES AND EDGES DEFINITION OF GRAPH GRAPH THEORY Prepared by Engr. JP Timola Reference: Discrete Math by Kenneth H. Rosen A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each

More information

Section 8.2 Graph Terminology. Undirected Graphs. Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between u and v.

Section 8.2 Graph Terminology. Undirected Graphs. Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between u and v. Section 8.2 Graph Terminology Undirected Graphs Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between u and v. The edge e connects u and v. The vertices u and v are

More information

Graphs. The ultimate data structure. graphs 1

Graphs. The ultimate data structure. graphs 1 Graphs The ultimate data structure graphs 1 Definition of graph Non-linear data structure consisting of nodes & links between them (like trees in this sense) Unlike trees, graph nodes may be completely

More information

Chapter 2 Graphs. 2.1 Definition of Graphs

Chapter 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 information

(5.2) 151 Math Exercises. Graph Terminology and Special Types of Graphs. Malek Zein AL-Abidin

(5.2) 151 Math Exercises. Graph Terminology and Special Types of Graphs. Malek Zein AL-Abidin King Saud University College of Science Department of Mathematics 151 Math Exercises (5.2) Graph Terminology and Special Types of Graphs Malek Zein AL-Abidin ه Basic Terminology First, we give some terminology

More information

CS 311 Discrete Math for Computer Science Dr. William C. Bulko. Graphs

CS 311 Discrete Math for Computer Science Dr. William C. Bulko. Graphs CS 311 Discrete Math for Computer Science Dr. William C. Bulko Graphs 2014 Definitions Definition: A graph G = (V,E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge

More information

Graph and Digraph Glossary

Graph and Digraph Glossary 1 of 15 31.1.2004 14:45 Graph and Digraph Glossary A B C D E F G H I-J K L M N O P-Q R S T U V W-Z Acyclic Graph A graph is acyclic if it contains no cycles. Adjacency Matrix A 0-1 square matrix whose

More information

Basics of Graph Theory

Basics of Graph Theory Basics of Graph Theory 1 Basic notions A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Simple graphs have their

More information

Graphs V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

Graphs V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} Graphs and Trees 1 Graphs (simple) graph G = (V, ) consists of V, a nonempty set of vertices and, a set of unordered pairs of distinct vertices called edges. xamples V={,,,,} ={ (,),(,),(,), (,),(,),(,)}

More information

Adjacent: Two distinct vertices u, v are adjacent if there is an edge with ends u, v. In this case we let uv denote such an edge.

Adjacent: Two distinct vertices u, v are adjacent if there is an edge with ends u, v. In this case we let uv denote such an edge. 1 Graph Basics What is a graph? Graph: a graph G consists of a set of vertices, denoted V (G), a set of edges, denoted E(G), and a relation called incidence so that each edge is incident with either one

More information

Graph Theory CS/Math231 Discrete Mathematics Spring2015

Graph 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 information

Foundations of Discrete Mathematics

Foundations of Discrete Mathematics Foundations of Discrete Mathematics Chapters 9 By Dr. Dalia M. Gil, Ph.D. Graphs Graphs are discrete structures consisting of vertices and edges that connect these vertices. Graphs A graph is a pair (V,

More information

A good picture is worth a thousand words

A good picture is worth a thousand words A good picture is worth a thousand words VCU, Department of Computer Science CMSC 3 Graphs Vojislav Kecman Expressive power is the first explanation for a success of graphs More claims for graphs come

More information

MATH 363 Final Wednesday, April 28. Final exam. You may use lemmas and theorems that were proven in class and on assignments unless stated otherwise.

MATH 363 Final Wednesday, April 28. Final exam. You may use lemmas and theorems that were proven in class and on assignments unless stated otherwise. Final exam This is a closed book exam. No calculators are allowed. Unless stated otherwise, justify all your steps. You may use lemmas and theorems that were proven in class and on assignments unless stated

More information

March 20/2003 Jayakanth Srinivasan,

March 20/2003 Jayakanth Srinivasan, Definition : A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Definition : In a multigraph G = (V, E) two or

More information

Combinatorics Summary Sheet for Exam 1 Material 2019

Combinatorics Summary Sheet for Exam 1 Material 2019 Combinatorics Summary Sheet for Exam 1 Material 2019 1 Graphs Graph An ordered three-tuple (V, E, F ) where V is a set representing the vertices, E is a set representing the edges, and F is a function

More information

Discrete mathematics II. - Graphs

Discrete 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 information

1 Digraphs. Definition 1

1 Digraphs. Definition 1 1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

More information

Graphs: Introduction. Ali Shokoufandeh, Department of Computer Science, Drexel University

Graphs: Introduction. Ali Shokoufandeh, Department of Computer Science, Drexel University Graphs: Introduction Ali Shokoufandeh, Department of Computer Science, Drexel University Overview of this talk Introduction: Notations and Definitions Graphs and Modeling Algorithmic Graph Theory and Combinatorial

More information

BACKGROUND: A BRIEF INTRODUCTION TO GRAPH THEORY

BACKGROUND: A BRIEF INTRODUCTION TO GRAPH THEORY BACKGROUND: A BRIEF INTRODUCTION TO GRAPH THEORY General definitions; Representations; Graph Traversals; Topological sort; Graphs definitions & representations Graph theory is a fundamental tool in sparse

More information

CS 220: Discrete Structures and their Applications. graphs zybooks chapter 10

CS 220: Discrete Structures and their Applications. graphs zybooks chapter 10 CS 220: Discrete Structures and their Applications graphs zybooks chapter 10 directed graphs A collection of vertices and directed edges What can this represent? undirected graphs A collection of vertices

More information

An Introduction to Graph Theory

An Introduction to Graph Theory An Introduction to Graph Theory CIS008-2 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 information

Chapter 11: Graphs and Trees. March 23, 2008

Chapter 11: Graphs and Trees. March 23, 2008 Chapter 11: Graphs and Trees March 23, 2008 Outline 1 11.1 Graphs: An Introduction 2 11.2 Paths and Circuits 3 11.3 Matrix Representations of Graphs 4 11.5 Trees Graphs: Basic Definitions Informally, a

More information

Chapter 9 Graph Algorithms

Chapter 9 Graph Algorithms Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 3 Definitions an undirected graph G = (V, E)

More information

11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions

11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions Introduction Chapter 9 Graph Algorithms graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 2 Definitions an undirected graph G = (V, E) is

More information

Math 170- Graph Theory Notes

Math 170- Graph Theory Notes 1 Math 170- Graph Theory Notes Michael Levet December 3, 2018 Notation: Let n be a positive integer. Denote [n] to be the set {1, 2,..., n}. So for example, [3] = {1, 2, 3}. To quote Bud Brown, Graph theory

More information

Graph Theory. ICT Theory Excerpt from various sources by Robert Pergl

Graph Theory. ICT Theory Excerpt from various sources by Robert Pergl Graph Theory ICT Theory Excerpt from various sources by Robert Pergl What can graphs model? Cost of wiring electronic components together. Shortest route between two cities. Finding the shortest distance

More information

CSC Intro to Intelligent Robotics, Spring Graphs

CSC Intro to Intelligent Robotics, Spring Graphs CSC 445 - Intro to Intelligent Robotics, Spring 2018 Graphs Graphs Definition: A graph G = (V, E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge has either one or two

More information

Characterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)

Characterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4) S-72.2420/T-79.5203 Basic Concepts 1 S-72.2420/T-79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G

More information

Inf 496/596 Topics in Informatics: Analysis of Social Network Data

Inf 496/596 Topics in Informatics: Analysis of Social Network Data Inf 496/596 Topics in Informatics: Analysis of Social Network Data Jagdish S. Gangolly Department of Informatics College of Computing & Information State University of New York at Albany Lecture 1B (Graphs)

More information

CMSC 380. Graph Terminology and Representation

CMSC 380. Graph Terminology and Representation CMSC 380 Graph Terminology and Representation GRAPH BASICS 2 Basic Graph Definitions n A graph G = (V,E) consists of a finite set of vertices, V, and a finite set of edges, E. n Each edge is a pair (v,w)

More information

Introduction to Graph Theory

Introduction 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 information

v V Question: How many edges are there in a graph with 10 vertices each of degree 6?

v 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 information

Graph Theory. Part of Texas Counties.

Graph 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 information

2. CONNECTIVITY Connectivity

2. CONNECTIVITY Connectivity 2. CONNECTIVITY 70 2. Connectivity 2.1. Connectivity. Definition 2.1.1. (1) A path in a graph G = (V, E) is a sequence of vertices v 0, v 1, v 2,..., v n such that {v i 1, v i } is an edge of G for i =

More information

CS 561, Lecture 9. Jared Saia University of New Mexico

CS 561, Lecture 9. Jared Saia University of New Mexico CS 561, Lecture 9 Jared Saia University of New Mexico Today s Outline Minimum Spanning Trees Safe Edge Theorem Kruskal and Prim s algorithms Graph Representation 1 Graph Definition A graph is a pair of

More information

Lecture 13. Reading: Weiss, Ch. 9, Ch 8 CSE 100, UCSD: LEC 13. Page 1 of 29

Lecture 13. Reading: Weiss, Ch. 9, Ch 8 CSE 100, UCSD: LEC 13. Page 1 of 29 Lecture 13 Connectedness in graphs Spanning trees in graphs Finding a minimal spanning tree Time costs of graph problems and NP-completeness Finding a minimal spanning tree: Prim s and Kruskal s algorithms

More information

Number Theory and Graph Theory

Number Theory and Graph Theory 1 Number Theory and Graph Theory Chapter 6 Basic concepts and definitions of graph theory By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com

More information

CS 441 Discrete Mathematics for CS Lecture 26. Graphs. CS 441 Discrete mathematics for CS. Final exam

CS 441 Discrete Mathematics for CS Lecture 26. Graphs. CS 441 Discrete mathematics for CS. Final exam CS 441 Discrete Mathematics for CS Lecture 26 Graphs Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Final exam Saturday, April 26, 2014 at 10:00-11:50am The same classroom as lectures The exam

More information

Minimum-Spanning-Tree problem. Minimum Spanning Trees (Forests) Minimum-Spanning-Tree problem

Minimum-Spanning-Tree problem. Minimum Spanning Trees (Forests) Minimum-Spanning-Tree problem Minimum Spanning Trees (Forests) Given an undirected graph G=(V,E) with each edge e having a weight w(e) : Find a subgraph T of G of minimum total weight s.t. every pair of vertices connected in G are

More information

Graphs - I CS 2110, Spring 2016

Graphs - I CS 2110, Spring 2016 Graphs - I CS 2110, Spring 2016 Announcements Reading: Chapter 28: Graphs Chapter 29: Graph Implementations These aren t the graphs we re interested in These aren t the graphs we re interested in This

More information

Theory of Computing. Lecture 10 MAS 714 Hartmut Klauck

Theory of Computing. Lecture 10 MAS 714 Hartmut Klauck Theory of Computing Lecture 10 MAS 714 Hartmut Klauck Seven Bridges of Königsberg Can one take a walk that crosses each bridge exactly once? Seven Bridges of Königsberg Model as a graph Is there a path

More information

Lecture 5: Graphs & their Representation

Lecture 5: Graphs & their Representation Lecture 5: Graphs & their Representation Why Do We Need Graphs Graph Algorithms: Many problems can be formulated as problems on graphs and can be solved with graph algorithms. To learn those graph algorithms,

More information

Number Theory and Graph Theory

Number Theory and Graph Theory 1 Number Theory and Graph Theory Chapter 7 Graph properties By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 2 Module-2: Eulerian

More information

Decreasing a key FIB-HEAP-DECREASE-KEY(,, ) 3.. NIL. 2. error new key is greater than current key 6. CASCADING-CUT(, )

Decreasing a key FIB-HEAP-DECREASE-KEY(,, ) 3.. NIL. 2. error new key is greater than current key 6. CASCADING-CUT(, ) Decreasing a key FIB-HEAP-DECREASE-KEY(,, ) 1. if >. 2. error new key is greater than current key 3.. 4.. 5. if NIL and.

More information

Graph Theory. Connectivity, Coloring, Matching. Arjun Suresh 1. 1 GATE Overflow

Graph 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 information

Chapter 9 Graph Algorithms

Chapter 9 Graph Algorithms Introduction graph theory useful in practice represent many real-life problems can be if not careful with data structures Chapter 9 Graph s 2 Definitions Definitions an undirected graph is a finite set

More information

Graphs. Terminology. Graphs & Breadth First Search (BFS) Extremely useful tool in modeling problems. Consist of: Vertices Edges

Graphs. Terminology. Graphs & Breadth First Search (BFS) Extremely useful tool in modeling problems. Consist of: Vertices Edges COMP Spring Graphs & BS / Slide Graphs Graphs & Breadth irst Search (BS) Extremely useful tool in modeling problems. Consist of: Vertices Edges Vertex A B D C E Edge Vertices can be considered sites or

More information

UNDIRECTED GRAPH: a set of vertices and a set of undirected edges each of which is associated with a set of one or two of these vertices.

UNDIRECTED GRAPH: a set of vertices and a set of undirected edges each of which is associated with a set of one or two of these vertices. Graphs 1 Graph: A graph G = (V, E) consists of a nonempty set of vertices (or nodes) V and a set of edges E. Each edge has either one or two vertices associated with it, called its endpoints. An edge is

More information

Information Science 2

Information Science 2 Information Science 2 - Applica(ons of Basic ata Structures- Week 03 College of Information Science and Engineering Ritsumeikan University Agenda l Week 02 review l Introduction to Graph Theory - Basic

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. An Introduction to Graph Theory

SCHOOL 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... Cut-Points and Bridges.

More information

Chapter 9 Graph Algorithms

Chapter 9 Graph Algorithms Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be if not careful with data structures 3 Definitions an undirected graph G = (V, E) is a

More information

CHAPTER 14 GRAPH ALGORITHMS ORD SFO LAX DFW

CHAPTER 14 GRAPH ALGORITHMS ORD SFO LAX DFW SFO ORD CHAPTER 14 GRAPH ALGORITHMS LAX DFW ACKNOWLEDGEMENT: THESE SLIDES ARE ADAPTED FROM SLIDES PROVIDED WITH DATA STRUCTURES AND ALGORITHMS IN JAVA, GOODRICH, TAMASSIA AND GOLDWASSER (WILEY 2016) GRAPH

More information

Outline. Introduction. Representations of Graphs Graph Traversals. Applications. Definitions and Basic Terminologies

Outline. 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 information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 8

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 8 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 8 An Introduction to Graphs Formulating a simple, precise specification of a computational problem is often a prerequisite

More information

CSC 8301 Design & Analysis of Algorithms: Kruskal s and Dijkstra s Algorithms

CSC 8301 Design & Analysis of Algorithms: Kruskal s and Dijkstra s Algorithms CSC 8301 Design & Analysis of Algorithms: Kruskal s and Dijkstra s Algorithms Professor Henry Carter Fall 2016 Recap Greedy algorithms iterate locally optimal choices to construct a globally optimal solution

More information

CPS 102: Discrete Mathematics. Quiz 3 Date: Wednesday November 30, Instructor: Bruce Maggs NAME: Prob # Score. Total 60

CPS 102: Discrete Mathematics. Quiz 3 Date: Wednesday November 30, Instructor: Bruce Maggs NAME: Prob # Score. Total 60 CPS 102: Discrete Mathematics Instructor: Bruce Maggs Quiz 3 Date: Wednesday November 30, 2011 NAME: Prob # Score Max Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1 Problem 1 [10 points] Find a minimum-cost

More information

22 Elementary Graph Algorithms. There are two standard ways to represent a

22 Elementary Graph Algorithms. There are two standard ways to represent a VI Graph Algorithms Elementary Graph Algorithms Minimum Spanning Trees Single-Source Shortest Paths All-Pairs Shortest Paths 22 Elementary Graph Algorithms There are two standard ways to represent a graph

More information

WUCT121. Discrete Mathematics. Graphs

WUCT121. 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 information

CSE 21: Mathematics for Algorithms and Systems Analysis

CSE 21: Mathematics for Algorithms and Systems Analysis CSE 21: Mathematics for Algorithms and Systems Analysis Week 10 Discussion David Lisuk June 4, 2014 David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, 2014 1 / 26 Agenda 1 Announcements

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 13. An Introduction to Graphs

Discrete Mathematics for CS Spring 2008 David Wagner Note 13. An Introduction to Graphs CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 13 An Introduction to Graphs Formulating a simple, precise specification of a computational problem is often a prerequisite to writing a

More information

22.1 Representations of graphs

22.1 Representations of graphs 22.1 Representations of graphs There are two standard ways to represent a (directed or undirected) graph G = (V,E), where V is the set of vertices (or nodes) and E is the set of edges (or links). Adjacency

More information

Algorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95

Algorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95 Algorithms: Graphs Amotz Bar-Noy CUNY Spring 2012 Amotz Bar-Noy (CUNY) Graphs Spring 2012 1 / 95 Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Amotz

More information

ROOT A node which doesn t have a parent. In the above tree. The Root is A. LEAF A node which doesn t have children is called leaf or Terminal node.

ROOT A node which doesn t have a parent. In the above tree. The Root is A. LEAF A node which doesn t have children is called leaf or Terminal node. UNIT III : DYNAMIC STORAGE MANAGEMENT Trees: Binary tree, Terminology, Representation, Traversals, Applications. Graph: Terminology, Representation, Traversals Applications - spanning trees, shortest path.

More information

Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 7

Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 7 CS 70 Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 7 An Introduction to Graphs A few centuries ago, residents of the city of Königsberg, Prussia were interested in a certain problem.

More information

Graphs and Network Flows IE411. Lecture 21. Dr. Ted Ralphs

Graphs and Network Flows IE411. Lecture 21. Dr. Ted Ralphs Graphs and Network Flows IE411 Lecture 21 Dr. Ted Ralphs IE411 Lecture 21 1 Combinatorial Optimization and Network Flows In general, most combinatorial optimization and integer programming problems are

More information

DS UNIT 4. Matoshri College of Engineering and Research Center Nasik Department of Computer Engineering Discrete Structutre UNIT - IV

DS 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 information

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))

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)) 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 information

Lecture 10. Elementary Graph Algorithm Minimum Spanning Trees

Lecture 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 information

Review: Graph Theory and Representation

Review: Graph Theory and Representation Review: Graph Theory and Representation Graph Algorithms Graphs and Theorems about Graphs Graph implementation Graph Algorithms Shortest paths Minimum spanning tree What can graphs model? Cost of wiring

More information

Algorithms. Graphs. Algorithms

Algorithms. 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 information

15 Graph Theory Counting the Number of Relations. 2. onto (surjective).

15 Graph Theory Counting the Number of Relations. 2. onto (surjective). 2. onto (surjective). You should convince yourself that if these two properties hold, then it is always going to be the case that f 1 is a function. In order to do this, you should remember the definition

More information

Math.3336: Discrete Mathematics. Chapter 10 Graph Theory

Math.3336: Discrete Mathematics. Chapter 10 Graph Theory Math.3336: Discrete Mathematics Chapter 10 Graph Theory Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall

More information

Solving Linear Recurrence Relations (8.2)

Solving Linear Recurrence Relations (8.2) EECS 203 Spring 2016 Lecture 18 Page 1 of 10 Review: Recurrence relations (Chapter 8) Last time we started in on recurrence relations. In computer science, one of the primary reasons we look at solving

More information

MAT 7003 : Mathematical Foundations. (for Software Engineering) J Paul Gibson, A207.

MAT 7003 : Mathematical Foundations. (for Software Engineering) J Paul Gibson, A207. MAT 7003 : Mathematical Foundations (for Software Engineering) J Paul Gibson, A207 paul.gibson@it-sudparis.eu http://www-public.it-sudparis.eu/~gibson/teaching/mat7003/ Graphs and Trees http://www-public.it-sudparis.eu/~gibson/teaching/mat7003/l2-graphsandtrees.pdf

More information

CS4800: Algorithms & Data Jonathan Ullman

CS4800: Algorithms & Data Jonathan Ullman CS4800: Algorithms & Data Jonathan Ullman Lecture 11: Graphs Graph Traversals: BFS Feb 16, 2018 What s Next What s Next Graph Algorithms: Graphs: Key Definitions, Properties, Representations Exploring

More information

CS DATA STRUCTURES AND ALGORITHMS

CS DATA STRUCTURES AND ALGORITHMS Computer Science and Engineering Third Semester CS1211 - DATA STRUCTURES AND ALGORITHMS UNIT-I - INTRODUCTION TO DATASTRUCTURES 1.Write down the definition of data structures? PART -A A data structure

More information

Course Introduction / Review of Fundamentals of Graph Theory

Course Introduction / Review of Fundamentals of Graph Theory Course Introduction / Review of Fundamentals of Graph Theory Hiroki Sayama sayama@binghamton.edu Rise of Network Science (From Barabasi 2010) 2 Network models Many discrete parts involved Classic mean-field

More information

Graph. Vertex. edge. Directed Graph. Undirected Graph

Graph. Vertex. edge. Directed Graph. Undirected Graph Module : Graphs Dr. Natarajan Meghanathan Professor of Computer Science Jackson State University Jackson, MS E-mail: natarajan.meghanathan@jsums.edu Graph Graph is a data structure that is a collection

More information

Binary Relations McGraw-Hill Education

Binary 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 information

Graph Theory S 1 I 2 I 1 S 2 I 1 I 2

Graph 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 information

Trees. Arash Rafiey. 20 October, 2015

Trees. Arash Rafiey. 20 October, 2015 20 October, 2015 Definition Let G = (V, E) be a loop-free undirected graph. G is called a tree if G is connected and contains no cycle. Definition Let G = (V, E) be a loop-free undirected graph. G is called

More information

Ordinary Differential Equation (ODE)

Ordinary 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 information

Chapter 1 Graph Theory

Chapter 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 information

Simple graph Complete graph K 7. Non- connected graph

Simple 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 information

CSE 417: Algorithms and Computational Complexity. 3.1 Basic Definitions and Applications. Goals. Chapter 3. Winter 2012 Graphs and Graph Algorithms

CSE 417: Algorithms and Computational Complexity. 3.1 Basic Definitions and Applications. Goals. Chapter 3. Winter 2012 Graphs and Graph Algorithms Chapter 3 CSE 417: Algorithms and Computational Complexity Graphs Reading: 3.1-3.6 Winter 2012 Graphs and Graph Algorithms Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.

More information

ECE 242 HOMEWORK 5. In the figure below, one can go from node B to A but not from A to B.

ECE 242 HOMEWORK 5. In the figure below, one can go from node B to A but not from A to B. ECE 242 HOMEWORK 5 Question 1: Define the following terms. For lines with multiple terms,differentiate between the terms. Also draw a figure illustrating each term. (a) Directed graph and undirected graph

More information