Lecture 14: Graph Theory I Discrete Mathematical Structures: Theory and Applications

10 Discrete Mathematical Structures: Theory and Applications 2 Learning Objectives  Learn the basic properties of graph theory  Learn about walks, trails, paths, circuits, and cycles in a graph  Explore how graphs are represented in computer memory  Learn about Euler and Hamilton circuits  Learn about isomorphism of graphs  Explore various graph algorithms  Examine planar graphs and graph coloring

10 Discrete Mathematical Structures: Theory and Applications 3 Graph Definitions and Notation  The Königsberg bridge problem is as follows: Starting at one land area, is it possible to walk across all of the bridges exactly once and return to the starting land area?  In 1736, Euler represented the Königsberg bridge problem as a graph, as shown in Figure 10.1(b), and answered the question in the negative.

10 Discrete Mathematical Structures: Theory and Applications 4 Graph Definitions and Notation  Consider the following problem related to an old children’s game. Using a pencil, can each of the diagrams in Figure 10.2 be traced, satisfying the following conditions? 1.The tracing must start at point A and come back to point A. 2.While tracing the figure, the pencil cannot be lifted from the figure. 3.A line cannot be traced twice.  As in geometry, each of points A, B, and C is called a vertex of the graph and the line joining two vertices is called an edge.

10 Discrete Mathematical Structures: Theory and Applications 5 Graph Definitions and Notation  The services are to be connected subject to the following condition: The pipes must be laid so that they do not cross each other. Consider three distinct points, A, B, C, as three houses and three other distinct points, W, T, and E, which represent the water source, the telephone connection point, and the electricity connection point.  Suppose there are three houses, which are to be connected to three services — water, telephone, and electricity — by means of underground pipelines.

10 Discrete Mathematical Structures: Theory and Applications 6 Graph Definitions and Notation  Try to join W, T, and E with each of A, B, and C by drawing lines (they may not be straight lines) so that no two lines intersect each other (see Figure 10.3).  This is known as the three utilities problem.

10 Discrete Mathematical Structures: Theory and Applications 7 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 8 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 9 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 10 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 11 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 12 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 13 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 14 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 15 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 16 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 17 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 18 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 19 Graph Definitions and Notation

10 Discrete Mathematical Structures: Theory and Applications 20 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 21 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 22 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 23 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 24 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 25 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 26 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 27 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 28 Walks, Paths, and Cycles

10 Discrete Mathematical Structures: Theory and Applications 29 Walks, Paths, and Cycles