Planar graph duality. E.g. the facts that
- A set of edges forms a connected spanning subgraph of a planar graph G if and only if the complementary set of edges forms an acyclic subgraph of the dual, and vice versa.
- Since spanning trees are just connected acyclic subgraphs, it follows that a subgraph is a tree if and only if its complement is dual to a tree. Euler's formula follows immediately.
- The edges that are not in the minimum spanning tree of a planar graph G are the duals of the edges that are in the maximum spanning tree of its dual.
- A planar graph is bipartite if and only if its dual is Eulerian.
- A planar graph is 3-connected (polyhedral) if and only if its dual is.
- The graphic matroid of a planar graph is the dual of the graphic matroid of the dual graph. Planar graphs are the only graphs for which the dual of the graphic matroid is also graphic.
- A directed planar graph is acyclic if and only if its dual graph (with the dual edges oriented 90 degrees clockwise from the primal ones) is strongly connected.