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Chapter 5: The Mathematics of Getting Around Section 3: Graph Concepts and Terminology Cont’d Tuesday, February 28, 2012 ◮ Example 5: Using the graph below, answer the following questions. A B b E b b b Z (a.) Give the vertex set V . V = {A, B, C, D, E, X, Y, Z} (b.) Give the edge set E . E = {AB, AD, AE, BE, CE, DE, XY, XY, Y Z} (d.) What vertices are adjacent to E? A, B, C, and D (e.) What edges are adjacent to AB? AD, AE, and BE b D Figure 5 vertex A B C D E X Y Z degree 3 2 1 2 4 2 3 1 b C b (c.) List the degrees of each vertex. X b Y ◮ Example 6: Consider the graph with V = {A, B, C, D, E} and E = {AC, AE, BD, BE, CA, CD, CE, DE}. Without drawing a picture of the graph, (a.) list all the vertices adjacent to D. B, D, E (b.) list all the edges adjacent to BD. BE, CD, DE (c.) find the degree of D. Since there are three vertices adjacent to D, then there are three edges meeting at vertex D, so deg(D) = 3. Check your answers by drawing the graph. Circuits and Paths A circuit is a trip around part or all of the graph that begins and ends at the same vertex and travels each edge at most once. A path is a trip around part or all of the graph that begins and ends at different vertices and travels each edge at most once. We can describe a path or circuit by listing the vertices in the order they are traveled. The length of a path or circuit is the number of edges in the path or circuit. ◮ Example 7: Using Figure 4 below, answer the following questions. Figure 4 (a.) What is the length of the path ABED? 3 (b.) Find two paths of length 3 from G to C. GFBC, GFBC (c.) Is ABCBAD a path? No, because you travel the edge AB more than once. (d.) Find all circuits of length 1. EE (e.) Find all circuits of length 2. BCB (f.) How many paths are there from D to B? 4; DEB, DEEB, DACB, DACB , DAB (g.) How many paths are there from F to G? 1; FG (h.) How many paths are there from D to G? 11; DEBFG, DEEBFG, DCABFG, DCABFG, DABFG, DABCBFG, DABCBFG, DEBCBFG, DEBCBFG, DEEBCBFG, DEEBCBFG (i.) How do your answers change to parts f, g, and h if we add the edge GH? Connected Graphs A graph is connected if it is in one piece. A disconnected graph is made up of connected components. Figure 4 Consider the figures to the right. Figure 4 shows a connected graph. Figure 6 shows a disconnected graph with 2 components. Figure 6 Figure 7 shows a disconnected graph with 3 components. Figure 7 ◮ Example 8: Give an example of a connected graph with six vertices such that every vertex has degree 2. Answers may vary. ◮ Example 9: Give an example of a disconnected graph with six vertices such that every vertex has degree 2. Answers may vary. Bridges An edge whose removal makes a connected graph disconnected is called a bridge. ◮ Example 10: Find all of the bridges in the graph shown in Figure 4. BF, FG, FH ◮ Example 11: Given an example of a connected graph with six vertices such that every edge is a bridge and the graph has exactly two vertices of degree 1. ◮ Example 12: If G is a connected graph with no bridges, how many vertices of degree 1 can G have? Why? None, the single edge at a vertex of degree must be a bridge because when this edge is removed, the vertex becomes isolated. Euler Paths and Euler Circuits An Euler path in a connected graph is a bath that travels through all of the edges of the graph. An Euler circuit in a connected graph is a circuit that travels through all of the edges of the graph. (a) (b) Figure 8. Figures for Example 13 and Example 14 ◮ Example 13: Find an Euler path in Figure 8(A). DABCEACDE ◮ Example 14: Find an Euler circuit in Figure 8(B). DABCEACDEFD ◮ Example 15: Consider the following game. You are given N vertices and are required to build a graph by adding edges connecting these vertices. Each time you add an edge you must pay $1. You can stop when the graph is connected. Develop a strategy that will cost you the least amount of money. What is the minimum amount of money you need to build the graph? (Give your answer in terms of N .)