List the cut vertices and bridges in the appropriate

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bridges, and blocks of G. List the cut-vertices and bridges in the appropriate places, and provide carefully labelled sketches of the blocks. Cut-vertices: Bridge(s): Block(s): _________________________________________________________________ 3. (10 pts.) Below, provide a proof by induction on the order of the graph G that every nontrivial connected graph G has a spanning tree. [Hint: If the order of the graph G is at least 3, Theorem 1.10 implies that G has a vertex v with G - v connected.]
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TEST2/MAD3305 Page 3 of 4 _________________________________________________________________ 5. (10 pts.) Apply Kruskal’s algorithm to find a minimum spanning tree in the weighted graph below. When you do this, list the edges in the order that you select them from left to right. What is the weight w(T) of your minimum spanning tree T? _________________________________________________________________ 6. (15 pts.) (a) Suppose G 1 and G 2 are nontrivial graphs. What does it mean mathematically to say that G 1 and G 2 are isomorphic?? [This is really a request for the definition!] (b) Sketch two graphs G and H that have the degree sequence s: 2, 2, 2, 2, 2, 2 and have the same order and size, but are not isomorphic. Explain briefly how one can readily see that the graphs are not isomorphic. (c) Explicitly realize C 5 and its complement below. [You may provide carefully labelled sketches.] Next, explicitly define an isomorphism from C 5 to its complement that reveals that C 5 is self-complementary.
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TEST2/MAD3305 Page 4 of 4 _________________________________________________________________ 7. (10 pts.) Find a minimum spanning tree for the weighted graph below by using only Prim’s algorithm and starting with the vertex g. When you do this, list the edges in the order that you select them from left to right. What is the weight w(T) of your minimum spanning tree T? _________________________________________________________________ 8. (15 pts.) (a) If G is a nontrivial graph, how is κ (G), the vertex connectivity of G, defined? (b) If G is a nontrivial graph, it is not true generally that if v is an arbitrary vertex of G, then either κ (G - v) = κ (G) - 1 or κ (G - v) = κ (G). Give a simple example of a connected graph G illustrating this. [A carefully labelled drawing with a brief explanation will provide an appropriate answer.] (c) Despite the example above, if G is a nontrivial graph and v is a vertex of G, κ (G - v) κ (G) - 1. Provide the simple proof for this.
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