Minimum-Area Drawings of Plane 3-Trees

A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The area of such a drawing is the area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊ 2n 3 − 1⌋ × 2⌈ n 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3 ⌋×⌊ 2(n−1) 3 ⌋ for plane graphs.

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