 Dual graph

For a graph with a cellular embedding on a surface, the dual has one vertex for each face and an edge crossing each original edge between its adjacent faces. Each edge is counted, including a loop when the two incident faces coincide. The planar dual graph is the case of a planar embedding.

In graph theory, a dual graph is a construction that relates to a planar graph. To understand dual graphs, it's important to start with the concept of a planar graph itself. A planar graph is a graph that can be drawn on a plane without any edges crossing. ### Key Concepts of Dual Graphs 1. **Vertices of the Dual Graph**: For every face (region) in the original planar graph, there is a corresponding vertex in the dual graph.