Graph theory

TEAM NAMES:

CSCI 4408/5408, Professor E. Gethner, Quiz 5, Spring 2023 Instructions: Be neat, write complete sentences, and show all of your work. The way you com- municate the solution to your answer is as important as the answer itself. This quiz is worth 80 points.

1. Disconnected Graph, 20 points. Give an example of a simple connected graph G on n vertices for which there exists a vertex v such that ω(G \ v) = n − 1 and each connected component of G \ v is an isolated vertex. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions.

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2. Spanning Trees, 40 points. Let G be the graph obtained from K5 by deleting two non- incident edges.

(a) (10 points) Assign the weights (1,1,2,2,3,3,4,4) to the edges of G so that the minimum spanning tree of G is unique. Call this graph G1. Be sure to explain in detail why the spanning tree is unique in this case. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions.

(b) (10 points) Next assign the weights (1,1,2,2,3,3,4,4) to the edges of G in another way so that the minimum spanning tree is not unique. Call this graph G2. Be sure to verify that the spanning tree is not unique in this case. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions.

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(c) (10 points) Run your graph G1 through Prim’s Algorithm and give a step-by-step ex- planation of the output. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions.

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(d) (10 points) Run your graph G2 through Kruskal’s Algorithm and give a step-by-step explanation of the output. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions.

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3. . Hamiltonian and Eulerian Cycles, 20 points

(a) (10 points) Display a Hamiltonian cycle in the icosahedral graph graph below.

(b) (10 points) Display an Eulerian cycle in the octahedral graph below. You will need to number the edges in order and with directional arrows to be able to describe the cycle.

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Extra workspace

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