Discrete Mathematics

Discrete Mathematics 6. GRAPHS Lecture 10 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com

Homework 9 Graph Gis given by the figure below. (a) List all possible paths from Ato C. (b) List all possible circuits. (c) Write down at least 4 cutsets of the graph. (d) Draw the subgraph G1 = {B,C,X,Y}. (e) Draw the complement of subgraph G1. Graph G

Solution of Homework 9 (a) All possible paths from A to C. (A,X,Y,C) and (A,X,B,Y,C) (b) All possible circuits. (B,X,Y,B) (c) At least 4 cutsets of the graph. {(A,Z)}, {(A,X)},{(C,Y)}, {(A,Z),(A,X)}, {(B,X),(B,Y)}, {(B,X),(X,Y)} Graph G

Solution of Homework 9 (d) Subgraph G1 = {B,C,X,Y}. (e) Complement of subgraph G1. Graph G

Bipartite Graph If the vertices of graph Gcan be separated into two subsets V1 and V2, such that every edge of G connects a vertex in V1 to a vertex in V2, then G is called a bipartite graph. Bipartite graph is denoted as as G(V1,V2). V1 V2

Bipartite Graph Is this graph a bipartite graph? Yes, because the vertices can be divided into two subsets V1 = {a,b,d} and V2 = {c,e,f,g}.

Isomorphic Graph Graphs that are actually identical but geometrically different are called isomorphic. Two graphs G1 and G2are 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 G1, then the corresponding edge e’ must be incident to vertex u’ and vertex v’ in G2. Two isomorphic graphs are identical graphs, different only in the naming of the vertices and edges or the geometrical representation only.

Isomorphic Graph • Graph (a) and graph (b) are isomorphic • Graph (a) and graph (c) are not isomorphic

Isomorphic Graph 2 isomorphic graphs 3 isomorphic graphs

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.

Planar Graph 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 Example of planar graph Example of non-planar graph

Plane Graph 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

Euler Path and Euler Circuit An Eulerpath 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.

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.

Euler Path and Euler Circuit 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. Graph (d) is an Eulerian graph. Graph (e) is neither semi-Eulerian nor Eulerian graph. Graph (f) is a semi-Eulerian graph.

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 Gcontains Eulercircuit if and only if it is connected and each of its vertices has even degree. In other words: An undirected graph Gis an Eulerian graph if and only if the degree of every vertex is even.

Euler Path and Euler Circuit Theorem: A directed graph Gcontains Euler path if and only if Gis connectedand for each vertex, the in-degree and out-degree are the same, excepttwo 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. Theorem: A directed graph Gcontains Eulercircuit if and only if Gis connected and for each vertex, the in-degree and out-degree are the same.

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.

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.

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.

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.

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.

Hamilton Path and Hamilton Circuit Example: Find a Hamilton circuit in the following graph.

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

Paths and Circuits 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. Graph (a) contains Euler path only. Graph (b) contains Euler path and Hamilton circuit. Graph (c) contains Euler circuit and Hamilton circuit.

Applications of Graphs Travelling salesman problem. Chinese postman problem. Graph coloring.

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).

Applications of TSP 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 Example: Determine the shortest Hamilton circuit in the following graph. Solution: There are 3 Hamilton circuits in the given graph above.

Applications of TSP P1 = (a, b, c, d, a) or (a, d, c, b, a) Total weight = 12 + 8 + 15 + 10 = 45 P2 = (a, b, d, c, a) or (a, c, d, b, a) Total weight = 12 + 9 + 15 + 5 = 41 P3 = (a, c, b, d, a) or (a, d, b, c, a) Total weight = 5 + 8 + 9 + 10 = 32 Shortest Hamilton circuit: P3

Chinese Postman Problem The problem was first discussed by Mei Ganin 1962. 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.

Chinese Postman Problem • 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 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 = 2 + 8 + 1 + 2 + 5 + 4 + 4 + 8 + 3 + 6 = 43.

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.

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

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.

Application of Graph Coloring 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 Map and corresponding graph representation Map Graph representation Graph coloring, 8 different colors Graph coloring, 4 different colors

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 itakes lecture j. Value 0 means student i does not take lecture j.

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)

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 Dcan be conducted together in one day. The exams of lectures B and Cshould be conducted in another day.

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.

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: TF1= {Amir, Budi, Yanti} TF2= {Budi, Hasan, Tommy} TF3= {Amir, Tommy, Yanti} TF4= {Hasan, Tommy, Yanti} TF5= {Amir, Budi} TF6= {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.

Discrete Mathematics