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Graph Theory
Any connected group of things can be represented as a graph: cities and roads, people and friendships, and more. Learn why an even number of people have an odd number of friends.
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Suppose a graph \(G\) is not planar. If we add an edge to \(G\) to connect two previously disconnected nodes to create the graph \(G'\), is \(G'\) planar?
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Suppose that there are three houses \(A, B, C\) and three utilities 1, 2, and 3 each of which needs to be connected by a wire to all three houses. Assuming the utilities and the houses are all points (nodes), is there a way to position them and the wires (edges) such that no two wires overlap?
Note: this is the same as asking whether \(K_{3,3}\) is a planar graph.
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Suppose \(G\) is a planar graph. If we create \(G'\) by adding a single edge to \(G\), is \(G'\) planar?
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