Lecture Notes On Planarity Testing And Construction Of Planar Embedding

A graph G is called “planar” if there is a way to draw it in the plane (e.g. on a piece of paper) such that there are no crossings among edges, except of course of the endpoints of edges which may coinside upon common vertices (shared by more than one edge). Given in another way: a graph G is planar if a way exists to draw it in the plane such that any of the plane's points are at most occupied by one vertex or one edge that passes through it. We seek a method which will first test a given graph for the property of planarity and then, if the graph is planar, will produce a representation of the apropriate drawing of the graph. This representation of the drawing which produces no crossings is called a “planar embedding” of the graph. It does not describe the length and shape of the edges of the graph nor the position of graph’s vertices. Instead, the topology of the graph is represented, which must be necessarily known, in order to produce a drawing. The final drawing of the graph is therefore one of the infinite instantiations of the topology that is described in the graph’s planar embedding, depending on the length, shape and position of edges and vertices respectively. The method we seek will not produce the actual planar drawing of the graph, but will compute the planar embedding. The way to use the planar embedding in order to produce an actual drawing with certain appearance and desired properties is another distinct problem, which we will not study here and highly depends on the application that uses the drawing. In figure 1.1, we depict an example of a graph subjected to planarity testing and a possible planar drawing of the graph, in order to show that the problem is not as easy as it seems from a first glance. Still, we remind that we do not seek the actual drawing, but only its topology.