Discrete Mathematics
Graph Theory
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?