Drawing planar graphs with few segments on a polynomial grid.

The visual complexity of a plane graph drawing is defined to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with 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. For trees, the grid size is $n\times n$. For 3-connected planar graphs and biconnected outerplanar graphs, we compute such drawings that are also monotone.