Perfect smooth orthogonal drawings

Smooth orthogonal drawings were recently introduced with the view of combining two different graph drawing approaches: Orthogonal drawings and Lombardi drawings. In this paper, we focus on perfect smooth orthogonal drawings in which each edge is made of either a rectilinear segment or a circular arc. We prove that every 3-planar graph admits a planar perfect smooth orthogonal drawing. If we relax planarity constraints, we show that every graph of maximum degree 4 admits a (non-planar) perfect smooth orthogonal drawing. We demonstrate that there exist infinitely many planar graphs that do not admit planar perfect smooth orthogonal drawings under the Kandinsky model. Finally, we introduce classes of graphs admitting perfect smooth orthogonal drawings of different styles and study relations between these classes.