(related to Chapter: Conditions for Planarity and Planarity Testing)
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.
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 $$|E|\le \frac{k}{k-2}(|V|-2).$$
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 $$|E|\le \frac{k}{k-2}(|F|-2).$$
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 $\deg (v)\le k+2.$
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 $\deg (f)\le k+2.$
