# Explanation: Examples of Dual Statements for Planar Graphs

The knowledge about the number of vertices, edges, and faces of a dual graph and the construction of dual graphs allows the dualization of many statements about connected planar graphs. We want to demonstrate this by the example of the proceeding necessary conditions for a biconnected graph to be planar.

### Original statement:

Let $k\ge 3$ be a positive integer and let $G(V,E)$ be a simple, biconnected, planar graph with $|E|$ edges and $|V|$ vertices and shortest cycle length $k.$ Then $$|E|\le \frac{k}{k-2}(|V|-2).$$

### Dualized statement

Let $k\ge 3$ be a positive integer and let $G(V,E)$ be a simple, biconnected, planar graph with $|E|$ edges and $|F|$ faces, and smallest vertex degree $k.$ Then $$|E|\le \frac{k}{k-2}(|F|-2).$$

### Original statement:

Let $k\ge 3$ be a positive integer and let $G(V,E)$ be a simple, biconnected, planar graph with $|E|$ edges and $|V|$ vertices, and shortest cycle length $k.$ Then $G$ contains a vertex $v$ of vertex degree $\deg (v)\le k+2.$

### Dualized statement

Let $k\ge 3$ be a positive integer and let $G(V,E)$ be a simple, biconnected, planar graph with $|E|$ edges and $|V|$ vertices and smallest vertex degree $k.$ Then $G$ contains a face $f$ of face degree $\deg (f)\le k+2.$

Github: ### References

#### Bibliography

1. Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000