samex2d - Sample Exam 2 EECS 203 Winter 2007 You have two...

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Sample Exam 2 EECS 203 Winter 2007 You have two hours to complete this exam. You may use any information you have written on an 8 . 5” × 11” sheet of paper (both sides), but no other information. Leave at least one seat between yourself and other students. Problem 1. Let G be a complete graph with 11 vertices. One or more of the following statements are true. Which ones are they? (a) The longest simple path is a circuit of length 11. (b) Removal of any one of the 11 vertices and all the edges incident to that vertex results in a complete graph with 10 vertices. (c) The adjacency matrix is an 11 × 11 matrix filled with 1’s. (d) G is isomorphic to all other complete graphs with 11 vertices. Problem 2. Consider the undirected graph H = ( V, E ) where V = { a, b, c, d, e } and E is the set {{ a, b } , { a, c } , { a, d } , { b, c } , { d, e }} . One or more of the following statements are true of H . Which ones are they? (a) H is connected. (b) The number of paths of length N from c to e is always equal to the number of paths of length N - 2 from c to a . (c) H is bipartite. (d) H is isomorphic to the undirected graph with vertex set { 1 , 2 , 3 , 4 , 5 } and edge set {{ 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 5 } , { 4 , 5 }} Problem 3. Suppose B is a wheel with 13 vertices. One or more of the following statements are true. Which ones are they? (a) Removing any one of the vertices results in a subgraph that is also a wheel. (b) The shortest path between any two distinct vertices is either 1 or 2. (c) Since the number of vertices of odd degree in an undirected graph must be even,
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