Drawing Series-Parallel Graphs on a Box

A box is a restricted portion of the three-dimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A box-drawing of a graph is a straight-line crossing-free drawing where vertices are located at integer grid points along the four lines. It is known that some planar graphs with tri-connected components do not admit a box-drawing. This paper shows that even structurally simpler planar graphs, namely series-parallel graphs, are not boxdrawable in general. On the positive side, it is proved that every series-parallel graph whose vertices have maximum degree at most three is box-drawable. A drawing algorithm is presented that computes a box drawing of a 3-planar series-parallel graph in optimal time and with optimal volume