The Cycle Space of an Embedded Graph III

This chapter presents an introduction to a cycle space of an embedded graph III. A basis for the cycle space of an embedded graph, first over Z₂ , the integers modulo 2, and then over an arbitrary field, was exhibited. The basis exhibited consisted of a maximal independent set of face boundaries together with certain fundamental cycles. The chapter gives a brief review of the main ideas and results. It also gives the description of a basis for the cycle space of an embedded graph over a commutative ring. The fact that the set of fundamental cycles forms a basis for Z. This can be proved in the usual way, the core of the argument being that no non-zero cycle can have as its support only edges from a tree.