R. Bar-Yehuda © www.cs.technion.ac.il/~cs234141 1 קומבינטוריקה למדעי - המחשב – הרצאה #16 EULER GRAPHS גרפים אויילרים מבוסס על הספר : S. Even, "Graph Algorithms",

Slides:

Advertisements

Similar presentations

Chapter 8 Topics in Graph Theory

Advertisements

Introduction to Graph Theory Instructor: Dr. Chaudhary Department of Computer Science Millersville University Reading Assignment Chapter 1.

EMIS 8374 Vertex Connectivity Updated 20 March 2008.

1 Lecture 5 (part 2) Graphs II Euler and Hamiltonian Path / Circuit Reading: Epp Chp 11.2, 11.3.

Lecture 21 Paths and Circuits CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine.

Section 14.1 Intro to Graph Theory. Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all.

Koenigsberg bridge problem It is the Pregel River divided Koenigsberg into four distinct sections. Seven bridges connected the four portions of Koenigsberg.

Graphs. Overview What is a graph? Some terminology Types of graph Implementing graphs (briefly) Some graph algorithms Graphs 2/18.

R. Bar-Yehuda © 1 Graph theory – תורת הגרפים 4.4 CATALAN NUMBERS מבוסס על הספר : S. Even, "Graph Algorithms", Computer.

R. Bar-Yehuda © 1 קומבינטוריקה למדעי - המחשב – הרצאה # DE BRUIJN SEQUENCES מבוסס על הספר : S. Even, "Graph Algorithms",

R. Bar-Yehuda © 1 קומבינטוריקה למדעי - המחשב – הרצאה #17 Chapter 2: TREES מבוסס על הספר : S. Even, "Graph Algorithms",

R. Bar-Yehuda © 1 Graph theory – תורת הגרפים 2.4 Directed Trees – עצים מכוונים מבוסס על הספר : S. Even, "Graph Algorithms",

R. Bar-Yehuda © 1 Graph theory – תורת הגרפים 2.6 THE NUMBER OF SPANNING TREE מבוסס על הספר : S. Even, "Graph Algorithms",

R. Bar-Yehuda © 1 קומבינטוריקה למדעי - המחשב – הרצאה #14 Graph theory – תורת הגרפים Chapter 1: PATHS IN GRAPHS – 1. מסלולים.

Discrete Structures Chapter 7B Graphs Nurul Amelina Nasharuddin Multimedia Department.

R. Bar-Yehuda © 1 Graph theory – תורת הגרפים 2.3 CAYLEY’S THEOREM – משפט קיילי מבוסס על הספר : S. Even, "Graph Algorithms",

Is C 6 2-critical? a). Yes b). No c). I have absolutely no idea.

CTIS 154 Discrete Mathematics II1 8.2 Paths and Cycles Kadir A. Peker.

4/17/2017 Section 8.5 Euler & Hamilton Paths ch8.5.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 5,Wednesday, September 10.

Graphs. Graph A “graph” is a collection of “nodes” that are connected to each other Graph Theory: This novel way of solving problems was invented by a.

Chapter 4 Graphs.

Chapter 11 Graphs and Trees This handout: Terminology of Graphs Eulerian Cycles.

R. Bar-Yehuda © 1 Graph theory – תורת הגרפים 4. ORDERED TREES 4.1 UNIQUELY DECIPHERABLE CODES מבוסס על הספר : S. Even,

The Bridge Obsession Problem By Vamshi Krishna Vedam.

Take a Tour with Euler Elementary Graph Theory – Euler Circuits and Hamiltonian Circuits Amro Mosaad – Middlesex County Academy.

Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.

CSE, IIT KGP Euler Graphs and Digraphs. CSE, IIT KGP Euler Circuit We use the term circuit as another name for closed trail.We use the term circuit as.

1 CS104 : Discrete Structures Chapter V Graph Theory.

5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life.

Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1.

Examples Euler Circuit Problems Unicursal Drawings Graph Theory

Chapter 5 Graphs  the puzzle of the seven bridge in the Königsberg,  on the Pregel.

Introduction to Graph Theory

Aim: What is an Euler Path and Circuit?

Graphs Edge(arc) Vertices can be even or odd or undirected (two-way) Edges can be directed (one-way) This graph is connected. Degree(order) = 3 Odd vertex.

Vertex-Edge Graphs Euler Paths Euler Circuits. The Seven Bridges of Konigsberg.

Graph theory and networks. Basic definitions  A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins.

Euler Paths and Circuits. The original problem A resident of Konigsberg wrote to Leonard Euler saying that a popular pastime for couples was to try.

1.Quiz 5 due tomorrow afternoon in E309 from 1pm to 4pm. 2.Homework grades will be based on ten graded homework assignments (dropping the lowest one).

MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles

Chapter 6: Graphs 6.1 Euler Circuits

 Quotient graph  Definition 13: Suppose G(V,E) is a graph and R is a equivalence relation on the set V. We construct the quotient graph G R in the follow.

1) Find and label the degree of each vertex in the graph.

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 14.1 Graphs, Paths, and Circuits.

Chap 7 Graph Def 1: Simple graph G=(V,E) V : nonempty set of vertices E : set of unordered pairs of distinct elements of V called edges Def 2: Multigraph.

1 GRAPH Learning Outcomes Students should be able to: Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using.

9.5 Euler and Hamilton graphs. 9.5: Euler and Hamilton paths Konigsberg problem.

Copyright © 2009 Pearson Education, Inc. Chapter 14 Section 1 – Slide Graph Theory Graphs, Paths & Circuits.

Euler Paths and Circuits

Konigsberg’s Seven Bridges

Discrete Math: Hamilton Circuits

Can you draw this picture without lifting up your pen/pencil?

Marina Kogan Sadetsky –

Connected Components Minimum Spanning Tree

תרגול 11 NP complete.

Graph Theory What is a graph?.

Euler and Hamilton Paths

Graphs G = (V, E) V are the vertices; E are the edges.

GRAPHS G=<V,E> Adjacent vertices Undirected graph

Chapter 5 Graphs the puzzle of the seven bridge in the Königsberg,

Euler circuit Theorem 1 If a graph G has an Eulerian path, then it must have exactly two odd vertices. Theorem 2 If a graph G has an Eulerian circuit,

Section 14.1 Graphs, Paths, and Circuits

Algorithms Lecture # 27 Dr. Sohail Aslam.

Graph Theory Relations, graphs

Warm Up – 3/19 - Wednesday Give the vertex set. Give the edge set.

Chapter 10 Graphs and Trees

Graph Theory: Euler Graphs and Digraphs

Presentation transcript:

R. Bar-Yehuda © 1 קומבינטוריקה למדעי - המחשב – הרצאה #16 EULER GRAPHS גרפים אויילרים מבוסס על הספר : S. Even, "Graph Algorithms", Computer Science Press, 1979 שקפים, ספר וחומר רלוונטי נוסף באתר הקורס : Slides, book and other related material at:

R. Bar-Yehuda © 2 לאונרד אוילר Leonhard Euler לאונרד אוילר Leonhard Euler 1707 נולד 1707 (באזל) 1723 מאסטר (פילוסופיה) קצין בצי הרוסי 1730 פרופסור (פיזיקה) 1733 ראש המחלקה למתימטיקה (St. Petersburg Academy of Science ) 866 ספרים ומאמרים 13 ילדים נחשב למתמטיקאי הפורה ביותר

R. Bar-Yehuda © 3 בעית טיול הגשרים בקניגסברג (Koenigsberg) בקנינגסברג היו 7 גשרים מעל נהר הפרגל (Pregel). הבעיה: כיצד ניתן לחצות את כל שבעת הגשרים מבלי לחצות גשר יותר מפעם אחת הפיתרון: ניתן לראשונה ע"י Euler ב-1736

R. Bar-Yehuda © 4 שלב א בפיתרון: ייצוג ע"י גרף

R. Bar-Yehuda © 5 EULER GRAPHS גרפים אויילרים יהי G=(V,E) גרף סופי לא מכוון. מסלול אויילרי (Euler path) ב-G הוא מסלול e 1,e 2,…, e L באורך L =|E| בו כל קשת מופיעה בדיוק פעם אחת במעגל אויילרי e 1 =e L

R. Bar-Yehuda © 6 EULER GRAPHS גרפים אויילרים גרף סופי לא-מכוון נקרא גרף אויילרי אם יש לו מסלול או מעגל אויילרי תרגיל רשות: האם הגרפים אויילרים?

R. Bar-Yehuda © EULER GRAPHS גרפים אויילרים Theorem 1.1: A finite (undirected) connected graph is an Euler graph if and only if exactly two vertices are of odd degree or all vertices are of even degree. In the latter case, every Euler path of the graph is a circuit, and in the former case, none is.

R. Bar-Yehuda © 8 גרפים מכוונים אויילרים Theorem 1.2: A finite digraph is an Euler digraph if any only if its underlying graph is connected and one of the following two conditions holds: 1.There is one vertex a s.t. d out (a) = d in (a) +1 and a vertex b s.t. d out (b)+1 = d in (b), for all other vertices v, d out (v) = d in (v). 2. For all vertices v, d out (v) = d in (v).