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# HW8_sol - Fall Semester 2008 CS300 Algorithms Solution#8...

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Fall Semester 2008 CS300 Algorithms Solution #8 1. (a) i. n - 1 ii. min { m, n } (b) i. True; By definition, a graph with connectivity 4 is 1-connected, 2-connected, 3-connected, and 4-connected, but not 5-connected. ii. False; Think about K 5 , a complete graph with 5 nodes. This graph is 1- connected, 2-connected, 3-connected, and 4-connected, but not 5-connected. Therefore, its connectivity is 4. 2. (a) Suppose that the condensation graph G 0 contains a cycle ( v 1 , v 2 , · · · , v m , v 1 ), m 2. Let u 1 be a vertex in the component S 1 and u m be a vertex in S m . Since G 0 contains both a v 1 - v m path and a v m - v 1 path, G has both a u 1 - u m path and a u m - u 1 path. Thus, u 1 and u m belong to the same strong component of G . This produces a contradiction. So, G 0 has no cycle. (b) Case 1: A newly added edge connects two vertices in the same component. Then G * 0 , the condensation graph of G * , has no cycle. Thus, the number of strongly connected components of G and that of G * are same.

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HW8_sol - Fall Semester 2008 CS300 Algorithms Solution#8...

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