Abstract. This article introduces a straight-line drawing algorithm for quadrangulations, in the family of the face-counting algorithms. It outputs in linear time a drawing on a regular W ×H grid such that W +H = n − 1 − ∆, where n is the number of vertices and ∆ is an explicit combinatorial parameter of the quadrangulation. 1

In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3-tree partition of the inner edges. Maximal planar bipartite graphs have a 2-tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NP-hardness of the k-tree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].

We show that there exist series-parallel graphs requiring Ω(n2 √ logn) area in any straightline or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. [Information Processing Letters, 2003]. Second, we show a family of series-parallel graphs requiring Ω(2 √ logn) width and Ω(2 √ logn) height in any straight-line or poly-line grid drawing. Combining the two results, the Ω(n2 √ logn) area lower bound is achieved. 2 1

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two transversal bipolar orientations. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of O ( √ n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (⌈n/2 ⌉ − 1) × ⌊n/2⌋.

We present binary labelings for the angles of quadrangulations and plane Laman graphs, which are in analogy with Schnyder labelings for triangulations [W. Schnyder, Proc. 1st ACM-SIAM Symposium on Discrete Algorithms, 1990] and imply a special tree decomposition for quadrangulations. In particular, we show how to embed quadrangulations on a 2-book, so that each page contains a non-crossing alternating tree.

Transversal structures on triangulations: a combinatorial study and straight-line drawings