However, it is known that not every graph has a Lombardi drawing and not every planar graph has a planar Lombardi drawing. Thus, not every graph can be drawn in this style, unless we relax the strict requirements of the definition.
We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs; we show that every graph has a smooth 2-Lombardi drawing and every planar graph has a smooth planar 3-Lombardi drawing.
For planar graphs, the big open question is whether every outerplanar Lombardi graph has a Lombardi drawing. We study planar Lombardi drawings for outerpaths, i.e., outerplanar graphs whose dual is a path. We show that every outerpath has an outerplanar Lombardi drawing and present a linear-time algorithm to construct it.