Smooth Orthogonal Drawings of Planar Graphs

In smooth orthogonal layouts of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axisaligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low edge complexity, that is, with few segments per edge. We say that a graph has smooth complexity k—for short, an SCk-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 SC2-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 SC1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that requires exponential area for an SC1-layout. Finally, we present an infinite family of biconnected 4-planar graphs that does not admit an SC1-layout.