Discrete Mathematics 6. GRAPHS Lecture 10 Dr.-Ing. Erwin Sitompul

10/2 Erwin SitompulDiscrete Mathematics Homework 9 Graph G is given by the figure below. (a)List all possible paths from A to C. (b) List all possible circuits. (c)Write down at least 4 cut sets of the graph. (d)Draw the subgraph G 1 = {B,C,X,Y}. (e)Draw the complement of subgraph G 1. Graph G

10/3 Erwin SitompulDiscrete Mathematics Solution of Homework 9 (a) All possible paths from A to C. Graph G (A,X,Y,C) and (A,X,B,Y,C) (b) All possible circuits. (B,X,Y,B) (c) At least 4 cut sets of the graph. {(A,Z) }, {(A,X) },{(C,Y)}, {(A,Z),(A,X)}, {(B,X),(B,Y)}, {(B,X),(X,Y)}

10/4 Erwin SitompulDiscrete Mathematics Solution of Homework 9 (d)Subgraph G 1 = {B,C,X,Y}. (e)Complement of subgraph G 1. Graph G

10/5 Erwin SitompulDiscrete Mathematics Bipartite Graph  If the vertices of graph G can be separated into two subsets V 1 and V 2, such that every edge of G connects a vertex in V 1 to a vertex in V 2, then G is called a bipartite graph.  Bipartite graph is denoted as as G(V 1,V 2 ). V1V1 V2V2

10/6 Erwin SitompulDiscrete Mathematics  Is this graph a bipartite graph? Bipartite Graph  Yes, because the vertices can be divided into two subsets V 1 = {a,b,d} and V 2 = {c,e,f,g}.

10/7 Erwin SitompulDiscrete Mathematics Isomorphic Graph  Graphs that are actually identical but geometrically different are called isomorphic.  Two graphs G 1 and G 2 are isomorphic if there is a one-to-one correspondence between vertices of the two graphs that preserves the adjacency relationship.  In other words, suppose the edge e is incident to vertex u and vertex v in G 1, then the corresponding edge e’ must be incident to vertex u’ and vertex v’ in G 2.  Two isomorphic graphs are identical graphs, different only in the naming of the vertices and edges or the geometrical representation only.

10/8 Erwin SitompulDiscrete Mathematics Isomorphic Graph Graph (a) and graph (b) are isomorphic Graph (a) and graph (c) are not isomorphic

10/9 Erwin SitompulDiscrete Mathematics Isomorphic Graph 2 isomorphic graphs 3 isomorphic graphs

10/10 Erwin SitompulDiscrete Mathematics Isomorphic Graph  From the definition of isomorphic graphs, it can be concluded that if two graphs are isomorphic, then both of them: 1. Have the same number of vertices. 2. Have the same number of edges. 3. Have the same number of vertices of each degree.  The 3 conditions listed above are necessary conditions, but not sufficient conditions.  Further visual inspection is required, as can be seen from the example below. The 3 conditions are met but both graphs are not isomorphic.

10/11 Erwin SitompulDiscrete Mathematics  A graph is called planar if it can be drawn in a plane without any edges crossing (where a crossing of edges is the intersection of the arcs representing them at a point other than their common vertices).  Such a drawing is called a planar representation of the graph.  If there is any edges crossing, then the graph called non- planar. Planar graph, the crossing edges can be rearranged and the graph can be redrawn without crossing Planar Graph

10/12 Erwin SitompulDiscrete Mathematics Example of non-planar graph Planar Graph Example of planar graph

10/13 Erwin SitompulDiscrete Mathematics  A planar graph which is drawn without any edges crossing is called a plane graph. Graph (a), (b), (c) are planar graphs Graph (b), (c) are plane graphs Plane Graph

10/14 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit  An Euler path in a graph is a path that contains every edge of the graph exactly once.  An Euler circuit in a graph is a circuit that contains every edge of a graph exactly once.  A graph that contains Euler path is also called semi-Eulerian graph.  A graph that contains Euler circuit is also called Eulerian graph.

10/15 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit Example:  Euler path in graph (a): 3, 1, 2, 3, 4, 1.  Euler path in graph (b): 1, 2, 4, 6, 2, 3, 6, 5, 1, 3, 5.  Euler circuit in graph (c): 1, 2, 3, 4, 7, 3, 5, 7, 6, 5, 2, 6, 1.  Graph (a) and (b) are semi-Eulerian graph.  Graph (c) is an Eulerian graph.

10/16 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit  Graph (d) is an Eulerian graph.  Graph (e) is neither semi-Eulerian nor Eulerian graph.  Graph (f) is a semi-Eulerian graph. Example:  Euler circuit in graph (d): a, c, f, e, c, b, d, e, a, d, f, b, a.  Graph (e) contains neither Euler path nor Euler circuit.  Graph (f) contains Euler path.

10/17 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit Theorem: An undirected graph G contains Euler path if and only if it is connected and has two vertices of odd degree or does not have any vertices of odd degree at all. Theorem: An undirected graph G contains Euler circuit if and only if it is connected and each of its vertices has even degree.  In other words: An undirected graph G is an Eulerian graph if and only if the degree of every vertex is even.

10/18 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit Theorem: A directed graph G contains Euler circuit if and only if G is connected and for each vertex, the in-degree and out-degree are the same. Theorem: A directed graph G contains Euler path if and only if G is connected and for each vertex, the in-degree and out-degree are the same, except two vertices, where the first vertex’s out- degree is one greater than the in-degree and the second vertex’s in-degree is one greater than the out-degree.

10/19 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit Example: (a) An Eulerian digraph: a, g, c, b, g, e, d, f, a. (b) A semi-Eulerian digraph: d, a, b, d, c, b. (c) A digraph, but neither Eulerian nor semi-Eulerian.

10/20 Erwin SitompulDiscrete Mathematics Euler Path and Euler Circuit Example : Is it possible to draw the graph below, by starting from any vertex, and without drawing any line twice? Solution : Yes, possible. All the vertices in the undirected graph above are of even degree. Therefore, the Euler circuit can be drawn. The graph is an Eulerian graph.

10/21 Erwin SitompulDiscrete Mathematics Bridges of Königsberg (Euler, 1736)  Can someone pass every bridge exactly once and come back the his/her original position? Solution : No, impossible. The degrees d(A) = 5, d(B) = 3, d(C) = 3, d(D) = 3  4 vertices of odd degree. The Euler circuit cannot be drawn.

10/22 Erwin SitompulDiscrete Mathematics Hamilton Path and Hamilton Circuit  A Hamilton path in a graph is a path that passes every vertex of the graph exactly once.  A Hamilton circuit in a graph is a circuit that passes every vertex of the graph exactly once, except one vertex which is the origin and (at the same time) the destination, is passed twice.  A graph that contains Hamilton path is also called semi- Hamiltonian graph.  A graph that contains Hamilton circuit is also called Hamiltonian graph.

10/23 Erwin SitompulDiscrete Mathematics Hamilton Path and Hamilton Circuit Example:  Graph (a) contains Hamilton paths: i.e., 3, 2, 1, 4.  Graph (b) contains Hamilton circuits: i.e., 1, 2, 3, 4, 1.  Graph (c) does not contain either Hamiltonian path or Hamiltonian circuit.

10/24 Erwin SitompulDiscrete Mathematics Example: Find a Hamilton circuit in the following graph. Hamilton Path and Hamilton Circuit

10/25 Erwin SitompulDiscrete Mathematics Theorem: A sufficient condition for a graph G with the number of vertices n  3 to be a Hamiltonian graph is that the degree of each vertex v in G to be at least n/2, or d(v)  n/2. Hamilton Path and Hamilton Circuit

10/26 Erwin SitompulDiscrete Mathematics Paths and Circuits  Graph (a) contains Euler path only.  Graph (b) contains Euler path and Hamilton circuit.  Graph (c) contains Euler circuit and Hamilton circuit.  A graph can contain Euler circuit/path and Hamilton circuit/path simultaneously.  A graph can also only contain Euler circuit/path or Hamilton circuit/path.

10/27 Erwin SitompulDiscrete Mathematics Applications of Graphs  Travelling salesman problem.  Chinese postman problem.  Graph coloring.

10/28 Erwin SitompulDiscrete Mathematics Travelling Salesman Problem (TSP)  For this problem, a number of cities and the distances between them are given.  Determine the shortest circuit that must be traveled by a salesman if he departs from a city of origin and stop by in each city exactly once and goes back to the city of origin.  This is a problem of how to find a Hamilton circuit with the minimum weight (distance).

10/29 Erwin SitompulDiscrete Mathematics  Mr. Postman collects the letters for mailboxes which are distributed in n locations in a certain town.  The robot arm fastens n bolts of a car in an assembly line.  Production process of n different products in one cycle. Applications of TSP

10/30 Erwin SitompulDiscrete Mathematics Applications of TSP Example: Determine the shortest Hamilton circuit in the following graph. Solution: There are 3 Hamilton circuits in the given graph above.

10/31 Erwin SitompulDiscrete Mathematics Applications of TSP  P 1 = (a, b, c, d, a) or (a, d, c, b, a) Total weight = = 45  P 2 = (a, b, d, c, a) or (a, c, d, b, a) Total weight = = 41  P 3 = (a, c, b, d, a) or (a, d, b, c, a) Total weight = = 32  Shortest Hamilton circuit: P 3

10/32 Erwin SitompulDiscrete Mathematics Chinese Postman Problem  The problem was first discussed by Mei Gan in  Problem: A postman will deliver the letters to the addresses in a part of a city. How should he plan the route of his journey so that he can pass each street exactly once and go back to the place where he starts his journey?  This is a problem of how to find an Euler circuit in a graph.

10/33 Erwin SitompulDiscrete Mathematics  If the graph of the problem is an Eulerian graph, then the Euler circuit can easily be found.  If the graph of the problem is not an Eulerian graph, then some edges in the graph must be passed more than once.  So, the postman must find a circuit that passes every street at least once with the shortest distance possible.  Chinese Postman Problem becomes: A postman will deliver the letters to the addresses in a part of a city. How should he plan his route so that:  The route has the shortest distance.  The postman passes every street at least once.  The postman goes back to his original position. Chinese Postman Problem

10/34 Erwin SitompulDiscrete Mathematics Example: Determine the best path that can be chosen by a postman so that he can pass every edge of the following graph at least once. Solution: The path that should be chosen by the postman is: A, B, C, D, E, F, C, E, B, F, A Weight = = 43. Chinese Postman Problem

10/35 Erwin SitompulDiscrete Mathematics Graph Coloring  A graph is colored in such a way that each vertex is given a color while two adjacent vertices may not have the same color.

10/36 Erwin SitompulDiscrete Mathematics Graph Coloring  Chromatic Number: the minimum number of colors required to color a graph.  Symbol:  (G), pronounced “k-eye”.  A graph G with chromatic number k is denoted as  (G) = k.  The graph below has  (G) = 3.

10/37 Erwin SitompulDiscrete Mathematics Application of Graph Coloring  Map Coloring  A map consists of a number of regions.  A map should be colored in such a way that two neighboring regions must have different colors.

10/38 Erwin SitompulDiscrete Mathematics  The regions are represented by the vertices, and the border between two neighboring regions is represented by an edge.  Coloring a region in a map means coloring the vertex in the corresponding graph.  Neighboring regions must have different colors  The color of every incident vertices must be different. Application of Graph Coloring

10/39 Erwin SitompulDiscrete Mathematics Application of Graph Coloring MapGraph representation Graph coloring, 8 different colors Graph coloring, 4 different colors Map and corresponding graph representation

10/40 Erwin SitompulDiscrete Mathematics Application of Graph Coloring  Scheduling  Suppose there are eight IE students batch 2009 (1, 2, …, 8) and five lectures available to be chosen (A, B, C, D, E).  The following table shows the matrix of five lectures and eight students.  Value 1 in a cell (i, j) means student i takes lecture j.  Value 0 means student i does not take lecture j.

10/41 Erwin SitompulDiscrete Mathematics Application of Graph Coloring  Problem: If in one day there may only be one exam, what is the minimum number of days required to schedule the exams such that every student can take his/her exams without any time conflicts?  Solution: vertex  lecture edge  there is at least one student who takes both lectures (which are connected by the edge)

10/42 Erwin SitompulDiscrete Mathematics Application of Graph Coloring Graph of exam schedule problem The result of graph coloring  The chromatic number is 2.  The exams of lectures A, E, and D can be conducted together in one day.  The exams of lectures B and C should be conducted in another day.

10/43 Erwin SitompulDiscrete Mathematics Homework 10, No.1 Take a look at the graphs (a), (b), and (c).  Determine whether each graph is an Eulerian graph, semi- Eulerian graph, Hamiltonian graph, or semi-Hamiltonian graph.  Give enough explanation to your answer.

10/44 Erwin SitompulDiscrete Mathematics Homework 10, No.2 A department has six task forces. Every task force conducts a routine monthly meeting. The member of the six task forces are: TF 1 = {Amir, Budi, Yanti} TF 2 = {Budi, Hasan, Tommy} TF 3 = {Amir, Tommy, Yanti} TF 4 = {Hasan, Tommy, Yanti} TF 5 = {Amir, Budi} TF 6 = {Budi, Tommy, Yanti} (a) What is the minimum number of time slots that must be allocated so that everyone that belong to more than one task force can attend the meetings that he/she must join without any time conflict? (b)Draw the graph that represents this problem and explain what do a vertex and an edge represent.