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 a vertex of $G$ appears at each pinchpoint of $P$; 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 determine five possible graphs with 7 vertices and 13 edges that could be self-dual embeddable in the pinched sphere, and we establish, by way of computer-powered methods, that such a self-embedding exists for exactly two of these five graphs.
"The Smallest Self-dual Embeddable Graphs in a Pseudosurface." Missouri J. Math. Sci. 30 (1) 85 - 92, May 2018. https://doi.org/10.35834/mjms/1534384958