Improving Curve Complexity Bounds for RAC Point-Set Embeddings of Trees

Determine whether, for every tree T and point set S, one can compute right-angle crossing (RAC) point-set embeddings with any prescribed number of edge crossings while achieving curve complexity strictly less than 9.

Background

Using their topological linear embeddings, the authors construct RAC point-set embeddings of trees with any prescribed number of crossings and curve complexity at most 9 (reducible to 6 when all points are collinear).

They pose the question of whether this curve complexity bound can be further lowered in the general setting.