The Smallest Self-dual Graphs in a Pseudosurface

A proper embedding of a graph G in a pseudosurface P is an embedding in which the regions of the complement of G in P are homeomorphic to discs and pinchpoints of P correspond to vertices in G; we say that a proper embedding of G in P is self dual if there exists an isomorphism from G to its topological dual. We show that each graph that has a possibility of being self-dual embeddable in a pseudosurface must have at least thirteen edges, and we establish other criteria that such a graph must satisfy. We show that there are five possible graphs that meet these criteria. Using the definition of an algebraic dual graph given by Abrams and Slilaty, we determine by way of computer-powered methods that exactly two of these five graphs are selfdual embeddable in the pinched sphere (the quotient of the sphere modulo the identification of two distinct points). We also utilize a surgery of Edmonds to produce self-dual embeddings of these graphs in the projective plane. We also determine that exactly one of the five graphs has a self-dual embedding in the projective plane and not in the pinched sphere.