Drawing Planar Graphs on Area

We contribute to an open problem in Graph Drawing [Brandenburg, F.J., D. Eppsein, M.T. Goodrich, S.G. Kobourov, G. Liotta and P. Mutzel, Selected open problems in graph drawing, in: Proc. Graph Drawing 2003, LNCS 2912, 2003, pp. 515–529] and improve the upper bound of the area of straight-line grid drawings of planar graphs to 4 3 n × 2 3 n . Our algorithm uses an improved version of the generic shift method [de Fraysseix, H., J. Pach and R. Pollack, How to draw a planar graph on a grid, Combinatorica 10 (1990), pp. 41–51.] with one shift for each good vertex and two shifts for each bad vertex. The key is the handling of “critical vertices”. gorithm runs in linear time. The drawing fits into an isosceles triangle with a horizontal base line and sides of slope ±1. Under this constraint the 8 9 n 2 bound is optimal for embedded planar graphs.