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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.

Planar Graphs


Is this graph planar?

Is this graph planar?

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?

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.

Suppose \(G\) is a planar graph. If we create \(G'\) by adding a single edge to \(G\), is \(G'\) planar?


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