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1 SSII Instructor: Benjamin Wilson Name: 1. Read each problem carefully and follow the instructions. 2. No credit will be given for correct answers without supporting work and/ or explanation. 3. You will not receive full credit unless your supporting work is clear, neat, and correct. 4. You may not use a calculator for this exam. 5. You do not need to simplify final answers to get full credit. 6. Clearly indicate final answers. 7. Sign the Honor Pledge: I pledge that I have neither given nor received unauthorized aid on this exam: Signature:
2 SSII (12 points) Model the map below as a graph making the landmasses vertices and the bridges and tunnels edges. Is there a way to travel around the map going over each bridge and through each tunnel exactly once? Explain.
3 SSII (16 points) Circle the letter for the correct answer(s) to the following questions. (i) A spanning tree of a network is a subgraph that is: (circle all that apply) A. Connected B. Doesn t have three edges meeting at one vertex C. Contains a circuit D. Doesn t contain a circuit E. Contains all vertices of the original graph (ii) Which of the following is the correct pronunciation of Euler? A. Eye-ler B. Yu-ler C. Oil-er D. Kruskal (iii) How many distinct Hamiltonian circuits does a complete graph with 12 vertices contain? A. 11! 2 B. 60 C. (12 + 2)! D. 12! 2 (iv) What is the degree of every vertex of K 8? A. 7 B. 8 C. 28 D. 7! (v) What is the sum of degrees of a graph with 10 edges? A. 10! B. 5 C. 10 D. 20 (vi) What do we call a graph without loops or multiple edges? A. Connected B. Simple C. Network D. Hamilton (vii) What is the redundancy of a network with 30 vertices and 45 edges? A. 15 B. 16 C. 29 D. 44 (viii) How many edges are there in K 10? A. 9 B. 10! C. 9! D. 45
4 SSII A traveling salesperson must fly to 5 different cities. The graph below gives the price of a plane ticket to fly between each of the cities which are labeled A-E. A $300 $50 $350 $150 E $450 $500 $200 B $250 $100 D $400 C (a) (3 points) Use the nearest-neighbor algorithm starting at vertex C to find a tour. (b) (1 point) Write the tour you found in (a) as a tour starting at B. (c) (1 point) Find the cost of the tour you found in part (a). CONTINUE ON NEXT PAGE
5 SSII (d) (10 points) Find the repetitive nearest-neighbor tour and give its cost. Write it as a tour starting at B.
6 SSII (15 points) You are in Phillips Hall (represented by A below) and you are told to go pick up lunch at three different restaurants in Chapel Hill (represented by B, C, and D below) for some friends before returning to Phillips. The times (in minutes) to get between each restaurant and Phillips are shown in the graph below. A 5 B D 9 C (a) How many distinct routes are possible for you to make the trip? (b) Find the cheapest-link tour and give its cost. (c) Use the brute-force algorithm to find the fastest route possible.
7 SSII (16 points) Decide if each of the following statements is true or false. If it is false, correct it by changing the underlined phrase. (i) The brute-force algorithm always finds an optimal solution to the traveling salesman problem. (ii) An Euler circuit is a circuit that visits all edges of a graph. (iii) If a graph is connected and has exactly two odd vertices then it has an Euler circuit. (iv) A complete graph with an even number of vertices will always contain an Euler circuit. (v) A complete graph will always contain a Hamilton circuit. (vi) A tree with N vertices has N + 1 edges. (vii) Kruskal s algorithm can always be used to find a Hamilton circuit. (viii) In a tree with 10 vertices there are 10 bridges.
8 SSII (8 points) (a) Eulerize the graph below. (b) For the following two graphs decide if an Euler path, Euler circuit or neither exists. Circle the answer and explain. Euler Circuit/ Euler Path/ Neither Explain: Euler Circuit/ Euler Path/ Neither Explain: 7. (8 points) Use Kruskal s algorithm to find a minimum cost spanning tree of the weighted network below. What is the cost?
9 SSII (10 points) Give an example of each of the following or explain why it is impossible. (i) A complete graph with 5 vertices. (ii) A graph with 5 even vertices and 5 odd vertices. (iii) A tree with a loop. (iv) A graph with 2 odd vertices that doesn t contain an Euler path. (v) A complete graph with 6 edges.
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Graph Theory Part of Texas Counties. We would like to visit each of the above counties, crossing each county only once, starting from Harris county. Is this possible? This problem can be modeled as a graph.