Editing Cubic graph (section)

==Topology and geometry==
Cubic graphs arise naturally in [[topology]] in several ways.  For example, the cubic graphs with ''2g-2'' vertices describe the different ways of cutting a [[surface]] of genus ''g ≥ 2'' into [[Pair of pants (mathematics) | pairs of pants]]. If one considers a [[Graph (discrete mathematics)|graph]] to be a 1-dimensional [[CW complex]], cubic graphs are ''[[Generic property|generic]]'' in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph.  Cubic graphs are also formed as the graphs of [[polyhedron|simple polyhedra]] in three dimensions, polyhedra such as the [[regular dodecahedron]] with the property that three faces meet at every vertex.

[[File:Graph-encoded map.svg|thumb|upright=1.8|Representation of a planar embedding as a graph-encoded map]]
An arbitrary [[graph embedding]] on a two-dimensional surface may be represented as a cubic graph structure known as a [[graph-encoded map]]. In this structure, each vertex of a cubic graph represents a [[Flag (geometry)|flag]] of the embedding, a mutually incident triple of a vertex, edge, and face of the surface. The three neighbors of each flag are the three flags that may be obtained from it by changing one of the members of this mutually incident triple and leaving the other two members unchanged.<ref>{{citation
 | last1 = Bonnington | first1 = C. Paul
 | last2 = Little | first2 = Charles H. C.
 | publisher = Springer-Verlag
 | title = The Foundations of Topological Graph Theory
 | year = 1995}}.</ref>