Every Large Point Set contains Many Collinear Points or an Empty Pentagon

Abel, Zachary; Ballinger, Brad; Bose, Prosenjit; Collette, Sébastien; Dujmović, Vida; Hurtado, Ferrán; Kominers, Scott Duke; Langerman, Stefan; Pór, Attila; Wood, David R.

doi:10.1007/s00373-010-0957-2

Cited by 14 publications

(31 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Conjecture 5 is true for t ≤ 5, but is open for t ≥ 6 or ≥ 4; see [1,10]. Given that, in general, Conjecture 5 is challenging, Jan Kára suggested the following weakening.…”

Section: Some Background Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Blocking Colored Point Sets

Aloupis

Ballinger

Collette

et al. 2012

Thirty Essays on Geometric Graph Theory

Self Cite

View full text Add to dashboard Cite

Abstract. This paper studies problems related to visibility among points in the plane. A point x blocks two points v and w if x is in the interior of the line segment vw. A set of points P is k-blocked if each point in P is assigned one of k colours, such that distinct points v, w ∈ P are assigned the same colour if and only if some other point in P blocks v and w. The focus of this paper is the conjecture that each k-blocked set has bounded size (as a function of k). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets {n1, n2, n3, n4} such that some 4-blocked set has exactly ni points in the i-th colour class. Amongst other results, for infinitely many values of k, we construct k-blocked sets with k 1.79... points.2000 Mathematics Subject Classification. 52C10 Erdős problems and related topics of discrete geometry, 05D10 Ramsey theory. This is the full version of an extended abstract to appear in the 26th European Workshop on Computational Geometry (EuroCG '10).

show abstract

“…Conjecture 5 is true for t ≤ 5, but is open for t ≥ 6 or ≥ 4; see [1,10]. Given that, in general, Conjecture 5 is challenging, Jan Kára suggested the following weakening.…”

Section: Some Background Motivationmentioning

confidence: 99%

“…, m }-midpointblocked point set, then P × Q is {n i m j : i ∈ [k], j ∈ [ ]}-midpoint-blocked. 1 If G is the visibility graph of some point set P ⊆ R d , then G is the visibility graph of some projection of P to R 2 (since a random projection of P to R 2 is occlusion-free with probability 1).…”

Section: Midpoint-blocked Point Setsmentioning

confidence: 99%

Blocking Colored Point Sets

Aloupis

Ballinger

Collette

et al. 2012

Thirty Essays on Geometric Graph Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this way, we obtain the vertices of the lower chain of D. For the upper chain, we place points on the unit circle with origin (0, 3) analogously. Note that line βn/2+1 passes through (1,3), so when picking rational points on the lower-right and lower-left quadrant of the second unit circle for the upper chain, the resulting point set is indeed the vertex set of a double chain in which the line through l 0 and u βn+1 is βn/2+1 . Finally, note that all slopes used in the construction have numerators and denominators that are polynomial in N .…”

Section: A1 Placing Points On Arcsmentioning

confidence: 99%

Flip Distance between Triangulations of a Simple Polygon is NP-Complete

Aichholzer

Mulzer

Pilz

2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study.We show that computing the flip distance between two triangulations of a simple polygon is NP-hard. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

show abstract

“…He also listed old and new examples of PP-sets. Abel et al [1] proved that for any fixed k, any sufficiently large PP-set contains k points on a line. Their bound on the size of the set was more than doubly-exponential.…”

Section: The Pentagon Propertymentioning

confidence: 99%

On planar point sets with the pentagon property

Cibulka

Kynčl

Valtr

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

Motivated by recent papers of Eppstein, Abel et al. and Barát et al., and by a question of Wood, we investigate properties of planar point sets with no 5-hole (no empty convex pentagon). We answer a question of Wood by showing that the visibility graph of a finite point set with no 5-hole may contain a clique of arbitrary size. This is in contrast with the previous examples of sets with no 5-hole, including the example of the (finite) square lattice. In our construction we use several equivalent local characterizations of (locally) finite planar point sets with no 5-hole which may be of independent interest. Our construction relies on a construction scheme which allows to derive new, non-trivial examples of sets with no 5-hole.

show abstract

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

Cited by 14 publications

References 35 publications

Blocking Colored Point Sets

Blocking Colored Point Sets

Flip Distance between Triangulations of a Simple Polygon is NP-Complete

On planar point sets with the pentagon property

Contact Info

Product

Resources

About