Compatibility of minimum-size and minimum-bend unbent collections

Determine whether every plane 4-graph admits an unbent collection whose size equals its unbent number and whose total number of bends equals its unbent bend number.

Background

The paper studies two objectives for unbent collections: minimizing the number of drawings, measured by the unbent number, and minimizing the total number of bends, measured by the unbent bend number. These objectives need not be automatically compatible, because a collection with the fewest drawings may require more total bends than a larger collection.

For plane triconnected cubic graphs, the paper proves that a minimum-bend collection can be computed with two drawings, and that two drawings are optimal in size. Whether this simultaneous optimality holds for every plane 4-graph is left unresolved.