Embedding Graph I: drawing & duality
In this section, graphs are permitted to have loops and multiple edges. Drawing If G is a graph, a drawing of G in the plane is a function f which assigns each vertex of G a distinct point in the plane, and assigns each edge uv of G a rectifiable curve with ends f (u), f (v) so that whenever the images of two edges share a vertex, it is either a vertex which is an end of both edges, or it is a cross. Planarity A graph G is planar if it may be drawn in the plane without crossings. A plane graph is a planar graph which is drawn in the plane without crossings. Faces If G is a plane graph, then the space obtained from the plane by removing all points in the image of G consists of finitely many connected components, each of which is called a face of G. Each face of G is bounded by a closed walk without repeated edges (not necessarily a cycle), and the length of a face is the length of this walk. An edge or vertex of G is incident with a face R if it appears in the walk which bounds R. Duality If G is a plane graph, then we may obtain a new plane graph G by creating a vertex in each face of G, and then joining two such vertices u, v by an edge called e whenever there is an edge e E(G) which borders the faces of G containing u and v. We say that the edge e in E(G ) corresponds to or is dual to the edge e E(G). Any graph obtained in this manner is called a dual of G. Observe that any dual graph G may be drawn in the plane with G so that each edge e ) E(G crosses the corresponding edge e E(G) at exactly one point, and so that these are the only points at which the drawing of G and the drawing of G meet. As suggested by the name, if G is a dual of G, then G is a dual of G . Thus, vertices of G correspond to faces of G , and faces of G correspond to vertices of G . Observation 5.1 Let G be a plane graph and let G be a dual of G. Then we have (i) If v V (G) and R is the corresponding face of G , then the degree of v is equal to the length of R. (ii) A set of edges C E(G) is the edge set of a cycle in G if and only if the corresponding set of edges C E(G ) is a bond of G . Proof: Part (i) follows immediately from the definitions. For part (ii), we begin by proving the "only if" direction. Let C be the edge set of a cycle in G. Then the image of C separates
2 the plane into two connected components, and this gives a partitio...