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

Abstract

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. Notation and DefinitionsIn 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 lef t(l), where right(l) and lef t(l) denote the open half-planes which are on the right side and on the left side of l, respectively (Figure 1). Let r 1 and r 2 be two rays emanating from the same point p. Then we denote by right(r 1 ) ∩ lef t(r 2 ) the open region that is swept by the ray being rotated clockwise around p from r 1 to r 2 (Figure 1). The open region lef t(r 1 ) ∩ right(r 2 ) is similarly defined. Then r 1 ∪ r 2 trisects the plane into three pieces: r 1 ∪ r 2 and two open regions right(r 1 ) ∩ lef t(r 2 ) and lef t(r 1 ) ∩ right(r 2 ). If the internal angle ∠r 1 pr 2 = ∠r 1 r 2 of right(r 1 ) ∩ lef t(r 2 ) is less than π, then we call right(r 1 ) ∩ lef t(r 2 ) the wedge defined by r 1 and r 2 , and denote it by wedge(r 1 r 2 ), wedge(r 2 r 1 ), wedge(r 1 pr 2 ) or wedge(r 2 pr 1 ). left(l)right ( Given a point p and a line l passing through p, let l * denote the line lying on l and having the opposite direction of l. We define the rays r and r * to be the two rays emanating from p and lying on the line l, while having the direction of l and l * respectively (Figure 1). Conversely, given a ray r, we can define r * , l and l * .For a set X of points in the plane, let conv(X) denote the convex hull of X, which is the smallest convex set containing X. For convenience, a region in the plane whose boundary consists of straightline segments is called a polygon even if it is an infinite region. For example, Figure 2

Long Alternating Paths in Bicolored Point Sets

“…We obtain many interesting new questions by considering colored point sets; see [KK04] for a survey. It is a well known mathematics contest problem to prove that between any set R of n red and any set B of n blue points in general position in the plane there is a noncrossing matching, i.e., a one-to-one correspondence between their elements so that the segments connecting the corresponding point pairs are pairwise disjoint.…”

Plane Bichromatic Trees of Low Degree

“…In [10] the authors mentioned that Kaneko proved Conjecture 1 for the case when |R| = |B|. However, we have not been able to find any written proof for this conjecture.…”

Hamiltonian Alternating Paths on Bicolored Double-Chains

“…In a more demanding version, the points and the vertices of the graph are colored and each vertex has to be placed in a point of the same color (see the survey [5] for further references). Interesting and non-trivial questions arise already if we want to embed a 2-colored path on a 2-colored point set.…”