Abstract. Fáry's theorem states that every plane graph can be drawn as a straightline drawing. A plane graph is a graph embedded in a plane without edge crossings. In this paper, we extend Fáry's theorem to non-planar graphs. More specifically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1-plane graphs for which every straight-line drawing has exponential area. To our best knowledge, this is the first result to extend Fáry's theorem to non-planar graphs.

We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain in R 3 specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold. Our algorithm encloses the input edges with a small buffer zone, a union of balls whose sizes are proportional to the local feature sizes at their centers. In the output mesh, the radius-edge ratio of the tetrahedra outside the buffer zone is bounded by a constant independent of the domain, while that of the tetrahedra inside the buffer zone is bounded by a constant depending on the smallest input angle. Furthermore, the output mesh is graded. Our work is the first that provides quality guarantees for Delaunay meshes in the presence of small input angles.

Map labeling encounters unique issues in the context of dynamic maps with continuous zooming and panning-an application with increasing practical importance. In consistent dynamic map labeling, distracting behavior such as popping and jumping is avoided. In our model a dynamic label placement is a continuous function that assigns a 2d-label to each scale. This defines a 3d-solid, with scale as the third dimension. To avoid popping, we truncate each solid to a single scale range, called its active range. This range corresponds to the interval of scales at which the label is visible. The active range optimization (ARO) problem is to select active ranges so that no two truncated solids overlap and the sum of the active ranges is maximized. We show that the ARO problem is NPcomplete, even for quite simple solid shapes, and we present constant-factor approximations for different variants of the problem.

Map labeling encounters unique issues in the context of dynamic maps with continuous zooming and panning-an application with increasing practical importance. In consistent dynamic map labeling, distracting behavior such as popping and jumping is avoided. In our model a dynamic label placement is a continuous function that assigns a 2d-label to each scale. This defines a 3d-solid, with scale as the third dimension. To avoid popping, we truncate each solid to a single scale range, called its active range. This range corresponds to the interval of scales at which the label is visible. The active range optimization (ARO) problem is to select active ranges so that no two truncated solids overlap and the sum of the active ranges is maximized. We show that the ARO problem is NPcomplete, even for quite simple solid shapes, and we present constant-factor approximations for different variants of the problem.

Annotating maps, graphs, and diagrams with pieces of text is an important step in information visualization that is usually referred to as label placement. We define nine label-placement models for labeling points with axis-parallel rectangles given a weight for each point. There are two groups: fixed-position models and slider models. We aim to maximize the weight sum of those points that receive a label.We first compare our models by giving bounds for the ratios between the weights of maximum-weight labelings in different models. Then we present algorithms for labeling n points with unit-height rectangles. We show how an O(n log n)-time factor-2 approximation algorithm and a PTAS for fixed-position models can be extended to handle the weighted case. Our main contribution is the first algorithm for weighted sliding labels. Its approximation factor is (2 + ε), it runs in O(n 2 /ε) time and uses O(n/ε) space. We show that other than for fixed-position models even the projection to one dimension remains NP-hard.For slider models we also investigate some special cases, namely (a) the number of different point weights is bounded, (b) all labels are unit squares, and (c) the ratio between maximum and minimum label height is bounded.