Visibility Drawings of Plane 3-Trees with Minimum Area Rahnuma Islam Nishat, Debajyoti Mondal and Md. Saidur Rahman Abstract. A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment such that the line segments use only grid points as their endpoints. The area of a visibility drawing is the area of the smallest rectangle on the grid which encloses the drawing. A minimum-area visibility drawing of a plane graph G is a visibility drawing of G where the area is the minimum among all pos- sible visibility drawings of G. The area minimization for grid visibility representation of planar graphs is NP-hard. However, the problem can be solved for a fixed planar embedding of a hier- archically planar graph in quadratic time. In this paper, we give a polynomial-time algorithm to obtain minimum-area visibility drawings of a plane 3-trees. Keywords. Visibility drawing, Plane 3-tree, Minimum layer, Minimum area. 1. Introduction Graph Drawing is a field of special attraction not only for the importance of aesthetic beauty of visual representations but also for its applicability in many areas. There are a number of drawing styles for drawing planar graphs and a “visibility drawing” is one of the most widely studied drawing styles among them. A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment. The vertical line segment representing an edge must connect points on the horizontal line segments representing the end vertices. Moreover, the horizontal line segments and the vertical line segments use only grid points as their ends. The area of a visibility drawing is the area of the smallest rectangle on the grid which encloses the drawing. A minimum-area visibility drawing of a plane graph G is a visibility drawing of G where the area is the minimum among all possible visibility drawings of G. In 1999, Lin and Eades gave a quadratic-time algorithm to obtain a minimum-area visibility drawing of a “hierarchically planar graph” with respect to a fixed planar embedding [10]. They also proved that the problem of finding a minimum-area visibility drawing of a planar graph is NP-hard in general. Many other interesting results on visibility drawings of plane graphs with small area are already present in the literature, which are the outcome of extensive study and research on visibility drawings [8, 9, 12]. Let G be a plane graph with n vertices. He and Zhang have given a linear-time algorithm to obtain a visibility drawing of G with height at most 2n 3 +2⌈  n/2⌉ and a visibility drawing with width at most 4n 3 +2⌈ √ n⌉ [7]. The known lower bound on the area requirement of visibility drawing is (⌊ 2n 3 ⌋) × (⌊ 4n 3 ⌋− 3) [13]. Researchers have also tried to obtain visibility drawings