In the dual graph, some constraints may be unnecessary.

The dual graph is a representation of how variables are constrained in the dual problem.

Some players find it easier to color vertices of the dual graph, as in the Four color theorem.

Each can be embedded in the orientable surface of genus 3, in which they form dual graphs.

The dual graph for this tiling is the complete tripartite graph K.

For example, problems whose primal or dual graphs are trees or forests can be solved in polynomial time.

In particular, a plane graph and its dual graph have the same medial graph.

Duality requires that every tie set has a dual cut set in the dual graph.

Consequently, the dual graph will not be cut into two parts and will not contain the required cut set.

The dual graph of the square lattice Z is also the square lattice.

However, if the graph is 3-connected, then Whitney showed that the embedding, and thus the dual graph, is unique.

The dual graph of this embedding is a symmetric 6-regular graph with 12 vertices and 36 edges.

A useful graph that is often associated with a triangulation of a polygon P is the so called dual graph.

With this embedding, the dual graph is the Petersen graph --- see hemi-dodecahedron.

They are the dual graphs of arrangements of lines in the hyperbolic plane that do not have three mutually-crossing lines.

More precisely, the dual graph contains a node for each dual variable and an edge for every constraint between them.

A graph obtained from the dual graph by removing some redundant edges is called a join graph.

Another method for finding out whether a constraint satisfaction problem has a join tree uses the primal graph of the problem, rather than the dual graph.

The algorithm can be used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation.

For 2-connected planar graphs, the pathwidth of the dual graph less than the pathwidth of the line graph.

The cut induced by such a partition in the dual graph corresponds to a Hamiltonian cycle in the primal graph.

Use the planar embedding to create an (undirected) graph T with the same vertex set as the dual graph of G.

Geometrically, the nerve is essentially a "dual complex" (in the sense of a dual graph, or Poincaré duality) for the covering .

If is planar, the dual of is the graphic matroid of the dual graph of .

The Tonnetz is the dual graph of Schoenberg's Chart of the regions, and of course vice versa.