We prove that every triangle-free planar graph is the intersection graph of a set of segments in the plane. Moreover, the segments can be chosen in only three directions (horizontal, vertical and oblique) and in such a way that no two segments cross, i.e., intersect in a common interior point. This particular class of intersection graphs is also known as contact graphs.

. Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recognition-complexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about bounded-ratio disk representations. We prove that the recognition of disk-intersection graphs (in the unbounded ratio case) is NP-hard. This result is proved in a more general setting of noncrossing arc-connected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unit-ball contact graphs is NP-hard in dimensions 3; 4, and 8 (24). 1 Introduction 1.1 Intersect...

Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a point-contact or a side-contact between the corresponding polygons. Specifically, we consider proportional contact representations, where pre-specified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2-segment graphs, which include maximal outerplanar graphs and partial 2-trees. 1

In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part we ask for representations by a dissections of a rectangle into squares. We