# Cycle double cover

Cycle double covers are a graph-theoretic abstraction that generalize the idea of a polyhedron. Like abstract polyhedra they don't intrinsically have spacial positioning but they have several key differences from abstract polyhedra.

## Motivation

Combinatorially, polygons are equivalent to cyclic graphs. Cyclic double covers take cyclic graphs and use them as faces to cover a skeleton creating a polyhedron.

## Definition

A cycle double cover is a simple graph G  along with a set F  of cycles in G  such that every edge of G  is in exactly 2 cycles of F .

## Comparison to related concepts

### Abstract polyhedra

All abstract polyhedra are cycle double covers. However the reverse is not true. Cycle double covers can be disconnected and have articulation points, while abstract polyhedra cannot. Even cycle double covers of connected graphs with no articulation points are not necessarily abstract polyhedra.

### Polygonal graph

Polygonal graphs are a subclass of cyclic double covers.