Cycle double cover

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A cycle double cover of the Petersen graph equivalent to the hemidodecahedron.

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

Abstract polyhedra[edit | edit source]

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[edit | edit source]

Polygonal graphs are a subclass of cyclic double covers.

Gallery[edit | edit source]

External links[edit | edit source]