Cartographic group

Definition

Let $X=(V,E)$ be a graph along with a combinatorial embedding in the plane. Then, the cartographic group or effective cartographic group of the combinatorial embedding is the subgroup of the permutation group on the ends of edges (the set of ends of edges is twice as large as the set of ends of edges, since for each edge we put two ends):

• A permutation that transposes the two ends of each edge, with each other
• A permutation that, for a fixed vertex, rotates all ends adjacent to it one step counter-clockwise

If the graph is connected, then the cartographic group acts transitively on the set of ends of edges. In essence, this means that we can go from any edge to any other by means of counter-clockwise rotations and by flipping the ends of edges.