Enumeration of acyclic and unicyclic nets with four types of self-duality

A net is a graph in which each point and line is given a sign. The point, line, and simple duals of a net are obtained by reversing the signs of the points, lines, or both. If a net possesses two of the three types of self-duality, it possesses all three and is said to be doubly self-dual. Enumeration formulas are derived for nets and point, line, simply, and doubly self-dual nets, whose underlying graphs are acyclic and unicyclic. The numbers are tabulated up to 12 points (24 for doubly self-dual nets) in each case.