# Bipartite and Series-Parallel Graphs Without Planar Lombardi Drawings

## TL;DR

This paper demonstrates that certain classes of bipartite, series-parallel, and apex-tree graphs cannot have planar Lombardi drawings, highlighting limitations in Lombardi graph visualization.

## Contribution

It identifies specific families of graphs that cannot be represented with planar Lombardi drawings, revealing fundamental constraints of Lombardi visualization methods.

## Key findings

- Planar bipartite graphs lack planar Lombardi drawings.
- Series-parallel graphs and apex-trees also lack such drawings.
- The results show inherent limitations of Lombardi drawings for these graph classes.

## Abstract

We find a family of planar bipartite graphs all of whose Lombardi drawings (drawings with circular arcs for edges, meeting at equal angles at the vertices) are nonplanar. We also find families of embedded series-parallel graphs and apex-trees (graphs formed by adding one vertex to a tree) for which there is no planar Lombardi drawing consistent with the given embedding.