Drawing Plane Graphs

Automatic aesthetic drawing of plane graphs has recently created intense interest due to its broad applications, and as a consequence, a number of drawing methods, such as the straight line drawing, convex drawing, orthogonal drawing, rectangular drawing and box-rectangular drawing, have come out [8,9,3,4,5,6,7, 10,11,14,16,23,29,33]. In this talk we survey the recent results on these drawings of plane graphs. The most typical method is a straight line drawing in which all edges of a plane graph are drawn as straight line segments without any edge-intersection, as illustrated in Fig. 1(a). Every plane graph has a straight line drawing [10, 31,34]. A straight line drawing of a plane graph G is called a grid drawing of G if the vertices of G are put on grid points of integer coordinates. The integer grid of size W × H consists of W + 1 vertical segments and H + 1 horizontal segments, and has a rectangular contour. It is known that every plane graph of n ≥ 3 vertices has a grid drawing on an (n − 2) × (n − 2) grid, and that such a grid drawing can be found in linear time [5,7,11,29]. It is also shown that, for each n ≥ 3, there exists a plane graph which needs a grid of size at least 2(n− 1)/3 × 2(n− 1)/3 for any grid drawing [6,11]. It has been conjectured that every plane graph has a grid drawing on a 2n/3 × 2n/3 grid, but the conjecture is still remained as an open problem. On the other hand, a restricted class of plane graphs has a more compact grid drawing [14,21]. Miura et al. [21] recently give a very simple algorithm which finds a grid drawing of any given 4-connected plane graph G on a W × H grid such that W = n/2 − 1 and H = n/2 in linear time if G has four or more vertices on the outer face. Since W = n/2 − 1 and H = n/2 , W +H ≤ n. Another typical method which often produces an aesthetic straight line drawing is a convex drawing, in which every face boundary is drawn as a convex polygon as illustrated in Fig. 1(b). Not every plane graph has a convex drawing, but every 3-connected plane graph has a convex drawing [33]. Thomassen obtained a necessary and sufficient condition for a plane graph to have a convex drawing [32]. Chiba et al. gave a linear algorithm to examine the condition and find a convex drawing [3,4]. A convex drawing is called a convex grid drawing if it is a grid drawing. Every 3-connected plane graph has a convex grid drawing on an (n−2)× (n−2) grid, and such a grid drawing can be found in linear time [5,30]. Miura et al. [22] recently give an algorithm which finds in linear time a convex grid drawing of any given 4-connected plane graph G on an integer grid such that W +H ≤ n− 1 if G has four or more vertices on the outer face boundary. Since W +H ≤ n− 1, W ×H ≤ (n− 1)/2 · (n− 1)/2 .