Partitioning a Planar Point Set into Empty Convex Polygons

Ding, Ren; Hosono, Kiyoshi; Urabe, Masatsugu; Xu, Changqing

doi:10.1007/978-3-540-44400-8_13

Cited by 5 publications

(6 citation statements)

References 2 publications

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“…In the case of G(n), the lower bound is improved to n+1 4 for any n, while the upper bound is improved to 9 34 n for any n and to 5n+1 19 for infinitely many n [3]. Thus, the current best bounds of both functions are given by the following result.…”

mentioning

confidence: 86%

“…(ii) a 3 lies in H(p 2 q; v 3 ): If γ (a 3 ; v l , q ) is not empty, we obtain H = {(v 1 v 2 p 1 ) 3 , (p 2 v 3 p 3 q) 4 , (v l p l pa 3 α 6 ) 5 } and R = γ (a 3 ; α 6 , v l ) for α 6 = α(a 3 ; v l , q ). See Fig.…”

Section: Thus We Obtainmentioning

confidence: 99%

“…iii) a3 lies in H(p1 p; v 2 ) ∩ H(p 2 q; v 2 ): We obtain H = {(v l v 1 p l ) 3 , (v 3 v 4 p 3 ) 3 , (pp 1 v 2 p 2 qa 3 ) 6 } and R = H(v l a 3 ; v 4 ) ∩ H(p l v 4 ; v l ). Since γ (v l ; p 3 , v 4 ) is empty, if we consider α = α(v l ; v 4 , p 4 ), we finally obtain H = {(p 1 v 2 p 2 ) 3 , (v 1 v 3 qp) 4 , (v l p l p 3 v 4 α) 5 } and R = H(v l α; v 5 ).The proof of the lemma is now complete since all the possible cases have been considered.…”

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confidence: 98%

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On the minimum number of mutually disjoint holes in planar point sets

Hosono¹

2015

Computational Geometry

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Let P be a set of n points in general position in the plane. In 1996, Urabe considered a partition of P into subsets S 1 ∪ · · · ∪ S l such that each S i forms a hole (or an empty convex polygon) of P and these holes are mutually disjoint. Let f (P ) be the minimum number of holes over all such partitions of P and F (n) = max{ f (P )} over all sets P of n points. Then the current best bounds are given by n+1 4 ≤ F (n) ≤ 5n 18 . In this paper, we prove that F (n) ≤ n 4 + 1.

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confidence: 86%

Section: Thus We Obtainmentioning

confidence: 99%

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confidence: 98%

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On the minimum number of mutually disjoint holes in planar point sets

Hosono¹

2015

Computational Geometry

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“…The proof uses a Ramsey type result for 11 points proposed by Aichholzer et al [1]. The most important case that remains to be settled is that of H (5,5). Urabe and Hosono [13] proved that 16 ≤ H(5, 5) ≤ 20, and later improved the lower bound to 17 [12].…”

Section: Remarks and Conclusionmentioning

confidence: 99%

“…14 . The upper bound bound was further improved to 9n 34 by Ding et al [5]. In [14], Urabe defined the function F k (n) = min S F k (S), where F k (S) is the maximum number of k-holes in a disjoint convex partition of S, and the the minimum being taken over all sets S of n points.…”

Section: Introductionmentioning

confidence: 99%

On the minimum size of a point set containing a 5-hole and a disjoint 4-hole

Bhattacharya

Das

2011

Studia Scientiarum Mathematicarum Hungarica

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Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.

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