We will also see that these two graphs are

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We will also see that these two graphs are essentially all nonplanar graphs!
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Embedding on the sphere Theorem 4. A graph is embeddable on the plane if and only if it is em- beddable on the sphere. Proof. Suppose that a graph G has an embedding ˜ G on the sphere. Choose a point x of the sphere not in ˜ G . The the image of ˜ G under the stereographic projection from x is an embedding of G on the plane. The converse can be proved similarly. 2 Duality Faces Definition 2. A plane graph G partitions the rest of the plane into a number of arcwise-connected open sets, called the faces of G . Each plane graph has exactly one unbounded face, called the outer face . face outerface The boundary of a face Definition 3. The boundary of a face f is the boundary of the open set f in the topological sense. Denote by ( f ) the boundary of a face f . The degree of f , deg( f ), is the number of edges in ( f ), with each bridge counted twice. A face is said to be incident with the vertices and edges in the boundary. Two faces are said to be adjacent if their boundaries have an edge in common. Any face can be made outer Theorem 5. Let G be a planar graph and f a face of some planar embed- ding of G . Then there exists a planar embedding of G whose outer face has the same boundary as f . Proof. Since G is planar, we may take an embedding ˜ G of G on the sphere. Denote by ˜ f the face of ˜ G corresponding to f . Let x be a point in the interior of ˜ f . Let π ( ˜ G ) be the image of ˜ G under stereographic projection from x . Then π ( ˜ G ) is a planar embedding of G which has the desired property.
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What does a face look like? Theorem 6. In a plane graph of order at least three with no loops or cut vertices, each face is bounded by a cycle (i.e. each boundary is a cycle). A proof of this result can be found in [J.A. Bondy and U.S.R. Murty, Graph Theory, 2008, Theorem 10.7]. Duals Definition 4. Given a plane graph G , define the dual of G , denoted by G * , as follows. Each face f of G corresponds to a vertex f * of G * , and each edge e of G corresponds to an edge e * of G * . Two vertices f * and g * are joined by the edge e * in G * if and only if their corresponding faces f and g are separated by the edge e in G . If e is a bridge, then f = g and so e * is a loop of G * . If e is a loop of G , then e * is a bridge of G * . G * is a planar graph, and we use the same notation G * for the ‘natural’ plane dual of the plane graph G . With this convention, we have G ** = G . # vertices of G * = # faces of G , # edges of G * = # edges of G deg G * ( f * ) = deg G ( f ) for all faces f of G Theorem 7. The dual of any plane graph is connected. Proof. Exercise! Theorem 8. If G is a plane graph with size m , then X f deg ( f ) = 2 m, where the sum is over all faces f of G .
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