Plane dual graph

For a connected plane graph G, denote its plane dual graph by G and define it on the set of faces of G with two faces being adjacent if they share an edge. Clearly, v(G ) = p(G) and p(G ) = v(G). 

Denote by Bundlem, m > 2, the plane graph with two vertices and m edges between them (so, m 2-gonal faces). The plane graph Bundlem, which is dual to m-gon, has the symmetry group Dmh = 7 (2,2, m) and it is a regular map, which is not a cell-complex. 

Figure 3.1 Smallest spherical, toroidal, Klein bottle and projective fullerenes. The first column lists the graphs drawn m the plane, the second the map on the appropriate surface and the third the dual on the same surface. The examples are (a) Dodecahedron (dual Icosahedron), (b) the Heawood graph (dual Ky), (c) a smallest Klein bottle polyhex (dual 3,3,3), and (d) the Petersen graph (dual Ke).

The notion of duality of plane graphs applies as well for (r, some applications in enumeration) and outer dual. They are always defined, but the resulting plane graph is not necessarily a (q, r)-polycycle. 

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