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The answer is n-1.
- CJ February 09, 2015But proving this is the trickier part.
This is one proof that I would have given. Convert the given problem to a graph problem. The players are treated as nodes and a comparison between 2 players is treated as an edge. Now, the problem boils down to graph connectivity. To have a connected graph, there should be a minimum of n-1 edges. If we have less than n-1 edges, it guarantees that the graph is not connected.
PS: While less than n-1 edges guarantees that the graph is not connected, greater than n-1 edges does not guarantee connectivity. I will leave it to you to figure out why.