Abstract. The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆-regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆-regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th

We consider straight-line drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges fromeachnodetoitschildrenfromonetofive, andtofivepatterns: straight, Y, ψ, X, and full. The ψ–drawings generalize hv- or strictly upward drawings to ternary trees. Weshowthatcompleteternarytreeshavea ψ–drawingonasquareofsize O(n 1.262) and general ternary trees can be drawn within O(n 1.631) area. Bothboundsareoptimal.Sub–quadraticboundsarealsoobtainedfor X– pattern drawings of complete tetra trees, and for full–pattern drawings of complete penta trees, which are 4–ary and 5–ary trees. These results parallel and complement the ones of Frati [8] for straight–line orthogonal drawings of ternary trees. Moreover, we provide an algorithm for compacted straight–line drawings of penta trees on the hexagonal grid, such that the direction of the edges from a node to its children is given by our patterns and these edges have the same length. However, drawing trees on a hexagonal grid within a prescribed area or with unit length edges is NP–hard.

Abstract. We show that every graph G with maximum degree three has a straight-line drawing in the plane using edges of at most five different slopes. Moreover, if every connected component of G has at least one vertex of degree less than three, then four directions suffice. 1

Abstract. Canonical ordering is an important tool in planar graph drawing and other applications. Although a linear-time algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. 1

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 height, width and area of such a drawing are respectively the height, width and 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 is the minimum. It is NP-complete to determine whether a plane graph G has a straight-line grid drawing with a given area or not. In this paper we give a polynomial-time algorithm for finding a minimum-area drawing of a plane 3-tree. Furthermore, we show a ⌊ 2n 3 −1⌋×2⌈n ⌉ lower bound for the area of a straight-line grid drawing of 3 a plane 3-tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3