Optimal 3D Angular Resolution for Low-Degree Graphs

We show that every graph of maximum degree three can be drawn without crossings in three dimensions with at most two bends per edge, and with 120◦ angles between all pairs of edge segments that meet at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5◦ angles, i. e., the angular resolution of the diamond lattice, between all pairs of edge segments that meet at a vertex or a bend. The angles in these drawings are the best possible given the degrees of the vertices. Submitted: July 2011 Reviewed: May 2012 Revised: January 2013 Accepted: February 2013 Final: February 2013 Published: March 2013 Article type: Regular paper Communicated by: H. Meijer This research was supported in part by the National Science Foundation under grants 0830403 and 1217322, by the Office of Naval Research under MURI grant N00014-08-1-1015, and by the German Research Foundation (DFG) under grant NO 899/1-1. E-mail addresses: [email protected] (David Eppstein) [email protected] (Maarten Löffler) [email protected] (Elena Mumford) [email protected] (Martin Nöllenburg) 174 Eppstein et al. Optimal 3D Angular Resolution for Low-Degree Graphs