Discrete Geometry on Red and Blue Points in the Plane – A Survey –

In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider balanced subdivision problems, geometric graph problems, graph embedding problems, Gallai-type problems and others. 1 Notation and Definitions In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider two disjoint sets R and B of red points and of blue points in the plane, respectively, such that no three points of R ∪ B lie on the same line. Throughout this paper, R and B always denote the sets mentioned above unless noted otherwise. We begin with some notation and definitions, which will be used throughout this paper. A (directed) line l trisects the plane into three pieces: l, right(l) and left(l), where right(l) and left(l) denote the open half-planes which are on the right side and on the left side of l, respectively (Figure 1). Let r1 and r2 be two rays emanating from the same point p. Then we denote by right(r1)∩ left(r2) the open region that is swept by the ray being rotated clockwise around p from r1 to r2 (Figure 1). The open region left(r1) ∩ right(r2) is similarly defined. Then r1 ∪ r2 trisects the plane into three pieces: r1 ∪ r2 and two open regions right(r1) ∩ left(r2) and left(r1) ∩ right(r2). If the internal angle ∠r1pr2 = ∠r1r2 of right(r1) ∩ left(r2) is less than π, then we call right(r1) ∩ left(r2) the wedge defined by r1 and r2, and denote it by wedge(r1r2), wedge(r2r1), wedge(r1pr2) or wedge(r2pr1).