# sol6 - MATH 239 ASSIGNMENT 6 Solutions 1. For each of the...

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MATH 239 ASSIGNMENT 6 Solutions 1. For each of the following graphs: c e f g h i a b G : H : e d f g h i a b c d (a) Find the breadth ±rst search tree, rooted at a , in which the vertices are added in alpha- betical order whenever there is a choice. (b) Prove or disprove that the graph is planar. Solution. (a) The breadth ±rst search trees are: d i h c f g e a BFST of G BFST of H f g a d c b e h i b (b) G is planar, since it can be redrawn as shown below (left). H has a subgraph which is an edge subdivision of K 3 , 3 , shown below (right). A subgraph of H G : c d i b a h g f e d f h a b c e g 2. (a) Let G be a connected graph. Suppose that the algorithm for growing a breadth ±rst search tree produces the exactly the same spanning tree no matter which vertex is chosen as the root. Prove that G is a tree. (b) Let G be a connected graph with at least 3 vertices. The depth of a breadth ±rst search tree is de±ned to be maximum of the levels of the vertices in the tree. Let d be a positive integer. Suppose that the algorithm for growing a breadth ±rst search tree produces a tree of depth d , no matter which vertex is chosen as the root. Prove that G does not contain a bridge.

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Solution. (a) Let v 1 ,... ,v p be the vertices of G , and let T i be the breadth Frst search tree rooted at v i . When the root is the active vertex in the algorithm for growing a breadth Frst search tree, every edge incident with the root gets added to the tree. Thus every edge incident with v i is in T i . Since every edge is incident with some vertex, it follows that every edge of
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## sol6 - MATH 239 ASSIGNMENT 6 Solutions 1. For each of the...

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