Mac Lane’s Theorem for 3-Graphs

In this chapter, we give our first characterisation of planar graphs. This characterisation is due to Mac Lane. The basic idea is that a 2-connected graph is planar if and only if there is a family of circuits enjoying properties that we intuitively expect the boundaries of faces to have. The work in this chapter is in the more general setting of 3-graphs. Mac Lane’s theorem is deduced as a corollary. We conclude the chapter by obtaining another criterion due to Whitney.