HW8 - 2 , ··· ,S p , and each edge ( v i ,v j ) in the...

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Fall Semester 2008 CS300 Algorithms Homework #8 (due 11/28) 1. The connectivity of G is the minimum size of a vertex set S such that G - S is disconnected or has only one vertex. A graph G is k - connected if its connectivity is at least k . (a) Give a connectivity value for the following graphs. i. A complete graph with n nodes ( K n ) ii. A bipartitle graph of bipartition X and Y of each of which has m and n nodes ( K m,n ) (b) Give a proof or a counterexample for each statement below. i. Every graph with connectivity 4 is 2-connected. ii. Every 3-connected graph has connectivity 3. 2. (a) Let S 1 ,S 2 , ··· ,S p be the strongly connected components of digraph G , then a con- densation graph G 0 = ( V 0 ,E 0 ) can be defined as follows: a set of vertices V 0 is composed of vertices v 1 ,v 2 , ··· ,v p which correspond to the strongly connected com- ponents S 1 ,S
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Unformatted text preview: 2 , ··· ,S p , and each edge ( v i ,v j ) in the set of edges E is defined when there is an edge between a vertex in S i and a vertex in S j . Now, prove the conden-sation graph G of digraph G defined above is a digraph with no cycle. (b) Let G * be a graph generated by adding a new edge to any digraph G . Discuss the difference between the number of strongly connected components of G and that of G * . 3. Find the all-pairs shortest path matrix for the weighted, directed graph represented by the following matrix W. Show the actions step by step (show each matrix value). W = 8 13 18 20 3 7 8 10 4 11 10 7 6 6 7 11 10 6 2 1 1...
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This note was uploaded on 02/04/2010 for the course COMPUTER S cs300 taught by Professor Unkown during the Spring '08 term at Korea Advanced Institute of Science and Technology.

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