By Claude Flament

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**Extra resources for Applications of graph theory to group structure**

**Example text**

15: Graphs for spanning tree problem. 2. Use the depth first search to find a spanning tree for the three graphs in problem 1. 5. Minimum Connector Problem 37 3. Use problem 2 to find a strongly connected orientation for all applicable graphs. 4. 15b that is different than the ones you found in problems 1 and 2. 5. Do two spanning trees of a graph always have a common edge? Prove or give a counterexample. 6. Show how to construct a rooted tree beginning with any vertex in a tree. Is it unique? 16.

3. Show that the sum of all scores is n(n2-1) if a tournament has n players. 4. 1. ) 5. Show that two isomorphic tournaments have the same score sequence. 6. Let T be a tournament with score sequence 3, 2, 2, 2, 1. Show there is a complete simple path starting from any vertex. Is this always true for strongly connected tournaments? 2. 7. In a recent presidential primary election, there were five candidates B, C, H, K, T. A committee of three was to choose the candidate to be supported by the local party.

10. This depth first search is another procedure that can be used in constructing a strongly connected orientation for a connected graph with no bridges. , consistent with an ordering of the vertices given by the order in which they are added, b, a, d, c, i, e, h, f, g. The remaining edges are oriented in the direction from the later vertices to the earlier ones. 13. 13: Orientation arismg from depth first search spanning tree. ) A tree, spanning or otherwise, always has a unique simple path between every pair of vertices.

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