One of the first algorithmic results in graph drawing was how to find a planar straight-line drawing such that vertices are at grid-points with polynomial coordinates. But not until 2007 was it proved that finding such a grid-drawing with optimal area is NP-hard, and the result was only for disconnected graphs. In this paper, we show that for graphs with bounded treewidth, we can find area-optimal planar straight-line drawings in one of the following two scenarios: (1) when faces have bounded degree and the planar embedding is fixed, or (2) when we want all faces to be drawn convex. We also give NP-hardness results to show that none of these restrictions can be dropped. In particular, finding area-minimal drawings is NP-hard for triangulated graphs minus one edge.
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