On the Total Number of Bends for Planar Octilinear Drawings

Abstract

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at $45^\circ$ and $-45^\circ$ line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. Since the problem of finding planar octilinear drawings with minimum number of bends is NP-hard, in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. on the slope number of planar graphs, we can derive an upper bound of $4n-10$ bends for planar graphs with $n$ vertices and maximum degree $8$. We considerably improve this general bound and corresponding previous ones for triconnected planar graphs of maximum degree $4$, $5$ and $6$. We also derive non-trivial lower bounds for these three classes of graphs by a technique inspired by the network flow formulation of Tamassia for computing bend optimal orthogonal drawings.