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hw3 - 22.2 a 1 The root of r is an articulation point of G...

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22.2 a) 1) The root of r π is an articulation point of G The root of G π has at least two children . - let r π be the root of G π . If r π has no child, the G has only one vertex r π . Thus r π is not articulation point of G according to definition. If r π has one child, all the vertices of G can be reachable through the child of r π . Since G is still connected graph even if r π is removed from G. Thus r π is not articulation point of G. 2) The root of G π has at least two children The root of G π is an articulation point of G. - Assume the root r π of G π has two child c1 and c2 and r π is not articulation point of G. Then there is a path between c1 and c2 which does not include r π , and we can reach c2 through c1. This means c2 is a sub-child node of c1, and it is contradict to our assumption that c1 and c2 are both children of r π . Thus r π is an articulation point of G. b) 1) v is an articulation point of G v has a child s such that there is no back edge from s or and descendant of s to a proper ancestor of v.
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