Conditions for Planar Graphs

As we have already explained in Section 1.4, a planar graph is a graph which can be drawn in the plane so that the edges have no intersections except at the vertices. We also gave a number of illustrations of planar graphs. In Section 1.5, we analyzed the problem of the three houses and the three wells, and explained why the corresponding graph could not be planar. The graph of the problem (Figure 1.16) can be drawn in many ways, as is possible for all graphs. When we say “the graph,” we mean any graph isomorphic to a particular graph describing the situation. Therefore, the statement “the graph in Figure 1.16 is not planar” means it has no planar isomorph. For instance, the vertices may be placed in a hexagon as in Figure 8.1. The intersection in the center is not a vertex; the edges should be considered to pass over each other at this point.

There is even a graph with only 5 vertices which is not planar—namely, the complete graph on 5 vertices (Figure 8.2). Why this graph is not planar may be made clear by reasoning similar to that used in Section 1.5 to show that the graph in Figure 8.1 (or Figure 1.16) is not planar. The vertices in any representation of the graph must lie on a cycle C in some order—say, abcdea. There is an edge eb, and in our planar graph we have the choice of putting it on the inside or the outside of C.