Euler's Formula, Proof 11: Ear Decomposition

A graph is two-edge-connected if removing any edge leaves a connected subgraph. Two-edge-connectivity is equivalent to the existence of an ear decomposition: a partition of the edges of the graph into a sequence of ears (simple paths and cycles), with the first ear being a single vertex; the start and end of each successive ear should be vertices occurring in previous ears, but all other vertices in an ear should be new. Such a decomposition can be found one ear at a time: start each ear by any unused edge e from an already-explored vertex, and continue by a shortest path back to another already-explored vertex (such a path must exist because e cannot disconnect the graph).

We can use such a decomposition in a proof of the Euler formula for polyhedra:

The graph $G$ of a polyhedron is two-edge-connected, since if we remove an edge $e$ we can still connect its endpoints by a path around the other side of one of the two faces of $G$ containing $e$.

Find an ear decomposition of $G$, and consider the process of forming $G$ by adding ears one at a time starting from a single vertex. Initially there is one vertex, one face, and no edges. Each new ear forms a path connecting two vertices on the boundary of a face, splitting the face in two; the path has one more edge than it has vertices. So if the ear has $k$ edges, its addition increases $V$ by $k-1$, $E$ by $k$, and $F$ by $1$, and the result follows by induction on the number of ears.

Ear decompositions have been especially useful in the design of parallel algorithms, since they are easier to find in parallel than are other structural decompositions of graphs such as depth first search trees. For an example of this see my work on recognizing series parallel graphs.

Proofs of Euler's Formula.
From the Geometry Junkyard, computational and recreational geometry pointers.
David Eppstein, Theory Group, ICS, UC Irvine.