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 breadth-ﬁrst 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 breadth-ﬁrst search tree T for which every non-tree edge joins vertices at equal levels. Prove that every cycle of G contains an even number of tree edges....
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This note was uploaded on 11/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
- Spring '09