Smooth Orthogonal Drawings of Planar Graphs

In \emph{smooth orthogonal layouts} of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axis-aligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low \emph{edge complexity}, that is, with few segments per edge. We say that a graph has \emph{smooth complexity} k---for short, an SC_k-layout---if it admits a smooth orthogonal drawing of edge complexity at most $k$. Our main result is that every 4-planar graph has an SC_2-layout. While our drawings may have super-polynomial area, we show that, for 3-planar graphs, cubic area suffices. Further, we show that every biconnected 4-outerplane graph admits an SC_1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that requires exponential area for an SC_1-layout. Finally, we present an infinite family of biconnected 4-planar graphs that does not admit an SC_1-layout.