gr-t2-a

Test2mad3305 page 4 of 4 7 10 pts find a minimum

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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? Beginning with vertex g, you should obtain edges in the following order: ge, ef, fb, ba, ed, dc. There is only one spanning tree, and its weight is w(T) = 31. You can check the edges, but not their order using Kruskal’s algorithm. _________________________________________________________________ 8. (15 pts.) (a) If G is a nontrivial graph, how is κ (G), the vertex connectivity of G, defined? If G is a complete graph of order n, then κ ( G )=n-1 . Otherwise, G has a vertex-cut. In this case, κ (G) = k where k is the cardinality of a minimum vertex-cut. (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 )-1o r 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.] Evidently, G K 1 +( K 1 K 2 ). Since G is connected and a is a cut-vertex of G, κ (G) = 1. Note, however,G-b K 3 , and thus κ ( G-b )=2 ,n o t0o r1 . (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. Proof: Let v be an arbitrary vertex of G. If G is a complete graph of order k, thenG-vi s complete of orderk-1 . Thus, the conclusion follows from definition of κ . So suppose G is not a complete graph. ThenG-vi sn o ta complete graph, too. Then either κ ( G-v ) ≥κ ( G )-1o r κ ( G-v )< κ
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