The graphs of polytopes with involutory automorphisms

Steinitz’ theorem states that a graph is the graph of a 3-dimensional convex polytope if and only if it is planar and 3-connected. Grunbaum has shown that Steinitz’ proof can be modified to characterize the graphs of polytopes that are centrally symmetric or have a plane of symmetry. We show how to modify Steinitz’ proof to take care of the remaining involutory case—polytopes that are symmetric about a line.