# Polygonal graph

**Polygonal graphs** are a combinatorial structures that generalize the idea of a regular polyhedron. Like abstract polyhedra they don't intrinsically have spacial positioning but they have several key differences from abstract regular polyhedra.

## Motivation[edit | edit source]

Abstract regular polygons can be seen as connected graphs with vertices of degree 2. Polygonal graphs attempt to generalize this idea to regular polyhedra. They thus start with a skeleton graph and add faces in the form of cycles on top.

## Definition[edit | edit source]

A polygonal graph is pair, (G , E ) consisting of a graph G and a set of cycles in G called "polygons", such that:

- G is regular (i.e. every vertex has the same degree)
- G is connected.
- G has girth m ≥ 3.
- Every polygon in E has size m .
- Every subgraph in G which is isomorphic to the star graph is a subgraph of exactly one polygon in E .

## Strict polygonal graphs[edit | edit source]

A polygonal graph is **strict** when its polygons consist of all the cycles of size m on its graph.

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### Abstract regular polyhedra[edit | edit source]

While there is wide overlap between polygonal graphs and abstract regular polyhedra there are some key differences:

- Abstract regular polyhedra allow faces to share more than 1 consecutive edges. For example dihedra are abstract regular polyhedra but not polygonal graphs. All such cases are considered degenerate.
- Abstract regular polyhedra allow digonal faces. For example hosohedra are abstract regular polyhedra but not polygonal graphs. All such cases are also considered degenerate.
- Polygonal graphs allow polyhedra which are not flag transitive so as they are equivelar. For example the Heawood map is a polygonal graph, but it is not considered regular as an abstract polytope.
- Polygonal graphs must have faces with size equal to the girth of the skeleton. Abstract regular polyhedra do not have this restriction.
- Polygonal graphs potentially allow more than two cycles to meet at an edge. Abstract polyhedra require dyadicity.

### Polyhedral maps[edit | edit source]

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## Bibliography[edit | edit source]

- Perkel, Manley (1979). "Bounding the valancy of polygonal graphs" (PDF).
*Canadian Journal of Math*.**31**(6): 1307–1321. - Perkel, Manley (1977).
*On finite groups acting on polygonal graphs*(PhD thesis). University of Michigan.

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