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# B k 33 is given below with partite sets u a b c and w

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(b) K 3,3 is given below with partite sets U = {a, b, c} and W = {A, B, C}. Give a connected bipartite subgraph of K 3,3 with the same partite sets but with U failing to be neighborly. Explain why your example satisfies the requirements. _________________________________________________________________ 12. (15 pts.) (a) State Kuratowski’s Theorem that characterizes planar graphs. (b) Without using Kuratowski’s Theorem, explain briefly how one can see that K 5 is not planar. (c) Without using Kuratowski’s Theorem, explain briefly how one can see that K 3,3 is not planar.

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MAD3305/Final Exam Page 7 of 8 _________________________________________________________________ 13. (25 pts.) (a) (5 pts.) What is a bridge? [Hint: Original definition please.] (b) (5 pts.) What is a cut-vertex?? [Definition, please.] (c) (15 pts.) For the graph G below, determine the cut-vertices, 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):
MAD3305/Final Exam Page 8 of 8 _________________________________________________________________ 14. (25 pts.) (a) (5 pts.) What is a network, N?? [Hint: It would be nice to see Gould’s definition.] (b) (5 pts.) What is a legal (or feasible) flow in a network N ? [ Hint: Definition. ] (c) (15 pts.) Obtain a maximum flow f in the network below, and verify the flow is a maximum by producing a set of vertices S that produces a minimum cut. Check that the total capacity of that cut is the same as the value of your max flow.
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b K 33 is given below with partite sets U a b c and W A B C...

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