Dual graph

Builds on Planar graph Graph embedding Multigraph

Definition

Let $G$ be a planar graph together with a fixed embedding in the plane, called a plane graph. The faces of $G$ are the connected components of the complement of the drawing in the plane. The dual $G^*$ is the multigraph, possibly with loops, defined as follows.

The vertices of $G^*$ are the faces of $G$ .
For each edge $e$ of $G$ there is one edge $e^*$ of $G^*$ .
The ends of $e^*$ are the faces on the two sides of $e$ ; if the same face lies on both sides, $e^*$ is a loop.

Remarks

Example: draw $K_4$ in the plane as a triangle with a central vertex joined to the corners. This drawing has $4$ faces, and every two of them share exactly one edge, so its dual is again $K_4$ . Non-example: $K_5$ has no dual in this sense, since it is not planar and so it has no plane embedding, which the setting requires. An edge $e$ is a bridge exactly when $e^*$ is a loop. If $G$ is connected, then $(G^*)^*$ is isomorphic to $G$ . A set of edges is a cycle of $G$ exactly when the dual edges form a minimal cut of $G^*$ . Different embeddings of the same planar graph can have non-isomorphic duals.

Reading path

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Graph definition
Graph embedding concept
Planar graph definition
Multigraph definition
Dual graph definition

Tag	`DPE8EDD9W7`
URI	`m2://DPE8EDD9W7`
Kind	definition
Writing	W0 stub
MSC	05C10
Defines	Dual graph
Uses	Planar graph, Graph embedding, Multigraph
Changed	2 November 2026

Dual graph

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here