Before revision:  In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a pathwith an arbitrary number of edges). A crossing-free straight-line drawing of a graph is called monotone if there is a monotone path between any pair of vertices with respect to some direction. We study drawings of trees, outerplanar graphs, and general planar graphs with few segments on a polynomial size grid.
After revision:  In particular, one object may represent multiple edges forming a path. Drawings of planar graphs with few segments on a polynomial size grid.