Planarity

Abstract

The two diagrams in Figure 8.1 represent the same graph, a K₄ with vertices a, b, c and d. As diagrams they are quite different: in the version K₄(1), the edges ac and bd cross; in K₄(2) there are no crossings. We shall refer to the two diagrams as different representations of the graph in the plane. The crossing number of a representation is the number of different pairs of edges that cross; the crossing number ν(G) of a graph G is the minimum number of crossings in any representation of G. A representation is called planar if it contains no crossings, and a planar graph is a graph that has a planar representation. In other words, a planar graph G is one for which ν(G) = 0. Figure 8.1 shows that ν(K₄) = 0.