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Unformatted text preview: nected component. No cycles means that no additional edges are given, yielding exactly | V | 1 edges. (b) Proof by contradiction. If the graph is not connected, then by de f nition there are at least two components. At least one of these components has i vertices with i or more edges (by the pigeonhole principle). Given i 1 edges to connect the component, the i th edge must then directly connect two of the vertices already connected through the other edges. The result is a cycle. Thus, to avoid a cycle, the graph must be connected. 69...
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- Fall '08