Unformatted text preview: E from G so that each component of the remaining graph is a tree containing exactly one vertex in S . Prove that E = 1 2  V ( G )  . 4. (8 marks) Consider the following graph G . 1 2 3 4 5 6 7 8 9 10 11 12 13 (a) Construct a breadthﬁrst search tree for G , taking the vertex labelled 1 as root. When considering the vertices adjacent to the vertex being examined, take them in increasing order of their labels. (b) Determine the set of vertices whose distance to vertex 1 is two. (c) Determine whether or not G is bipartite, and prove your assertion. 5. (5 marks) Suppose that a connected graph G has a breadthﬁrst search tree T for which every nontree edge joins vertices at equal levels. Prove that every cycle of G contains an even number of tree edges....
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 Spring '09
 M.PEI
 Math, Graph Theory, Planar graph, vertices, breadthfirst search tree, odddegree vertices

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