Bend-optimal orthogonal graph drawing in the general position model

We consider orthogonal drawings in the general position model, i.e., no two points share a coordinate. The drawings are also required to be bend minimal, i.e., each edge of a drawing in k dimensions has exactly one segment parallel to each coodinate direction that are glued together at k − 1 bends. We provide a precise description of the class of graphs that admit an orthogonal drawing in the general position model in the plane. The main tools for the proof are Eulerian orientations of graphs and discrete harmonic functions. The tools developed for the planar case can also be applied in higher dimensions. We discuss two-bend drawings in three dimensions and show that K2k+2 admits a k-bend drawing in k+ 1 dimensions. If we allow that a vertex is placed at infinity, we can draw K2k+3 with k bends in k + 1 dimensions. Mathematics Subject Classifications (2010) 05C62 (graph representations). ∗An extended abstract of this work was presented at EuroCG’12 with the title The graphs that can be drawn with one bend per edge. †Partially supported by DFG grant FE-340/7-2 and ESF EuroGIGA project GraDR. ‡Partially supported by project CE-ITI (GAČR P2020/12/G061) of the Czech Science Foundation.