Bipartite embeddings of trees in the plane

Abellanas, Manuel; García-López, Jesús; Hernández-Peñver, Gregorio; Noy, Marc; Ramos, Pedro

doi:10.1007/3-540-62495-3_33

Cited by 36 publications

(56 citation statements)

References 6 publications

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“…It is shown that if R and B are separated by a line and |R| = |B|, then there exists a non-crossing geometric alternating path that passes through all the points in R ∪ B [1]. On the other hand, as we show in Figure 4 (c), there exists a configuration consisting of six red points and seven blue points in the plane for which there exists no non-crossing geometric alternating path P 13 that passes through all these 13 points.…”

Section: Theorem 23 (The Equitable Subdivision Theoremmentioning

confidence: 99%

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

Kaneko

Kanō

2003

Algorithms and Combinatorics

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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

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Section: Theorem 23 (The Equitable Subdivision Theoremmentioning

confidence: 99%

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

Kaneko

Kanō

2003

Algorithms and Combinatorics

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“…Abellanas et al [2] proved that there exists a plane bichromatic tree on R ∪ B whose maximum vertex degree is O(k + log |B|). Kaneko [9] showed how to compute a plane bichromatic tree of maximum degree 3k.…”

Section: Previous Workmentioning

confidence: 99%

“…Abellanas et al [2] considered the problem of computing a low degree plane bichromatic tree on some restricted point sets. They proved that if R ∪ B is in convex position and |R| = k|B|, with k 1, then R ∪ B admits a plane bichromatic tree of maximum degree k + 2.…”

Section: Previous Workmentioning

confidence: 99%

“…In this paper we are interested in computing a plane bichromatic tree on R∪B whose maximum vertex degree is as small as possible. This problem was first mentioned by Abellanas et al [1] in the Graph Drawing Symposium in 1996:…”

Section: Introductionmentioning

confidence: 96%

“…For cases when |R| = |B| or |R| = |B|+1 it may not always be possible to compute a plane bichromatic tree of degree |R|−1 |B| + 1 = 2, i.e., a plane bichromatic path; see the example in the figure on the right which is borrowed from [2]; by adding one red point to the top red chain, an example for the case when |R| = |B| + 1 is obtained. In 1998, Kaneko [9] posed the following conjecture.…”

Section: Introductionmentioning

confidence: 99%

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Plane Bichromatic Trees of Low Degree

Biniaz

Bose

Maheshwari

et al. 2017

Discrete Comput Geom

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How to Embed a Path onto Two Sets of Points

2006

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Let R and B be two sets of points such that the points of R are colored red and the points of B are colored blue. Let P be a path such that |R| vertices of P are red and |B| vertices of P are blue. We study the problem of computing a crossing-free drawing of P such that each blue vertex is represented as a point of B and each red vertex of P is represented as a point of R. We show that such a drawing can always be realized by using at most one bend per edge.

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Bipartite embeddings of trees in the plane

Cited by 36 publications

References 6 publications

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

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

Plane Bichromatic Trees of Low Degree

How to Embed a Path onto Two Sets of Points

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