Rooted orientations: existence at high rooted connectivity

Ascertain whether, for each k, sufficiently highly rooted-connected graphs admit rooted k-connected orientations from a specified root; equivalently, determine a function r(k) such that every graph that is r(k)-rooted-connected from a root r has a rooted k-connected orientation.

Background

The paper proves the k=2 case: every rooted 6-connected graph admits a rooted 2-connected orientation, via rigidity-based methods. The general case is conjectural.

Rooted orientation problems blend connectivity augmentation with directionality constraints, and rigidity supplies novel tools to attack them.