... G is the minimum number of distinct edgeslopes in a straight-line drawing of G in the plane. We prove that for \Delta> = 5and all large n, there is a \Delta-regular n-vertex graph with slope-number at least n1-8+"\Delta +4. This is the best known lower bound on the slope-number of a graphwith bounded degree. We prove upper and lower bounds on the slope-numberof complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that theslope-number of interval graphs, cocomparability graphs, and AT-free graphs isat most a function of the maximum degree. We prove that graphs of boundeddegree and bounded treewidth have slope-number at most O(log n). Finallywe prove that every graph has a drawing with one bend per edge, in which thenumber of slopes is at most one more than the maximum degree. In a companionpaper, planar drawings of graphs with few slopes are also considered.

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include intersections of lattices with convex sets, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.

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

We consider embeddings of 3-regular graphs into 3-dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axis-parallel line) and such that no three points lie on the same axis-parallel line; we call a graph with such an embedding an xyz graph. We describe a correspondence between xyz graphs and face-colored embeddings of the graph onto two-dimensional manifolds, and we relate bipartiteness of the xyz graph to orientability of the underlying topological surface. Using this correspondence, we show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and three-connected, and that it is NP-complete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given graph is an xyz graph.

Abstract. We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique ” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005]. 1.

We give an algorithm to create orthogonal drawings of 3-connected 3-regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face.

Graph augmentation problems are motivated by network design, and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.

The distance-number of a graph G is the minimum number of distinct edge-lengths over all straight-line drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no K − 4-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that n-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in O(log n). To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as ∆ increases the existential lower bound on the distance-number of ∆-regular graphs tends to Ω(n0.864138). 1

We show that there exist triangulations that do not admit any planar straight-line drawing in which every face is an isosceles triangle, thus partially solving a question posed in [Demaine, Mitchell, O’Rourke – The Open Problems Project]. 2 1

A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all possible straight-line drawings of G. In this paper we prove that it is NP-complete to determine whether a plane graph G has a straight-line drawing with at most k segments, where k ≥ 3. We also prove that the problem of deciding whether a given partial drawing of G can be extended to a straight-line drawing with at most k segments is NP-complete, even when G is an outerplanar graph. Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph. 1