Realizability of vertex-orbit and arc-orbit pairs in nut graphs

Determine all pairs of nonnegative integers (r, k) for which there exists a nut graph with exactly r vertex orbits and k arc orbits.

Background

The paper completely characterizes the attainable pairs consisting of the number of vertex orbits and edge orbits of a nut graph: such a graph exists precisely when the number of edge orbits is at least one more than the number of vertex orbits. It also establishes infinite realizability for every admissible pair. The analogous realizability problem for vertex orbits and arc orbits is explicitly left unresolved.