Unformatted text preview: Here are two connected graphs. The single dot has one point, one region, and no lines, so since 1 + 1 = 0 + 2 it satisﬁes equation (1). The other has 4 points, 2 regions and 4 lines, so equation (1) correctly asserts 4 + 2 = 4 + 2. Your proof may use the following fact: Fact. If G is any connected graph, then at least one of the following is true: 1. G has only one point (and so has no lines and one region). 2. G has at least one edge with one free end, so that removing that edge and the free point will still leave a connected graph (with the same regions as before). 3. G has an edge which can be deleted, without removing any points, leaving a connected graph and merging two regions of the original graph into one region. In the example with 4 points, the upper line is an example of type 2. Any of the other three lines are examples of type 3: removing one of them will merge the region inside the triangle with the region outside, leaving only one region....
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- Fall '11
- Line segment, line segments, connected graph