---
tag: 1QHTD1DS5J
kind: concept
name: Dual graph
msc: [05C10]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_graph
same_as:
  wikidata: Q2294516
  wikipedia: Dual graph
---

## Summary

The dual graph of a graph drawn in the plane has one vertex for each face of the drawing.
It has one edge for each edge of the drawing, joining the two faces on its two sides.
Duality exchanges faces and vertices, and cycles and minimal cuts.
It depends on the chosen [[embedding|CG91WNMG2T]], not only on the abstract graph, and the same idea works for graphs embedded in other surfaces.

## Explanations

### L1 human

Think of a map of countries.
Put a capital in each country, and join two capitals by a road crossing each stretch of border that the two countries share.
The road network is the dual graph of the map of borders.
For example, the dual of the graph of a cube, which has $6$ faces, is the graph of an octahedron, which has $6$ vertices.