Self-dual graphs

We consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid self-duality. We show how these concepts are related with each other and with the connectivity of G. We use the geometry of self-dual polyhedra together with the structure of the cycle matroid to construct all self-dual graphs. 1.1. Forms of Self-duality. Given a planar graph G = (V, E), any regular embedding of the topological realization of G into the sphere partitions the sphere into regions called the faces of the embedding, and we write the embedded graph, called a map, as M = (V, E, F). G may have loops and parallel edges. Given a map M , we form the dual map, M * by placing a vertex f * in the center of each face f , and for each edge e of M bounding two faces f 1 and f 2 , we draw a dual edge e * connecting the vertices f * 1 and f * 2 and crossing e once transversely. Each vertex v of M will then correspond to a face v * of M * and we write M G has distinguishable embeddings, then G may have more than one dual graph, see Figure 1. In this example a portion of the map (V, E, F) is flipped over on a ¨ ¨ ¨ r r r d d d d e e e ¡ ¡ ¡ r r r ¨ ¨ ¨ ¡ ¡ ¡ d d d d e e e ¨ ¨ ¨ d d r r r r r d d¨¨ ¨ s s s s s s s s s s s s s c c c c c c c c e e s s s d d¨¨ ¨ c c c c c c c c r r r ¨ ¨ ¨ € € € € d d r r r Figure 1. separating set of two vertices to form (V, E, F). Such a move is called a Whitney flip, and the duals of (V, E, F) and (V, E, F) are said to differ by a Whitney twist. If the graph (V, E) is 3-connected, then there is a unique embedding in the plane and so the dual is determined by the graph alone. In general, an object is said …