Deciding whether the unbent number is two or three

Determine, for a given plane 4-graph, whether its unbent number is 2 or 3, thereby deciding efficiently whether two orthogonal drawings suffice or three are necessary.

Background

An unbent collection of a plane 4-graph is a collection of embedding-preserving orthogonal drawings in which every edge is straight in at least one drawing. The paper proves that every plane 4-graph has unbent number at most 3 and constructs graphs with unbent number 3, while also characterizing graphs with unbent number at most 2 through balanced 2-edge-colorings.

The unresolved algorithmic issue is therefore to distinguish efficiently between the cases in which two drawings suffice and those in which three drawings are required. A partial sufficient condition for unbent number at most 2 is established using Nash-Williams forest decompositions, but this does not yield a complete decision procedure for all plane 4-graphs.