Higher-order triangular-distance Delaunay graphs: Graph-theoretical properties

Biniaz, Ahmad; Maheshwari, Anil; Smid, Michiel

doi:10.1016/j.comgeo.2015.07.003

Cited by 8 publications

(13 citation statements)

References 20 publications

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“…The problem of computing a maximum matching in G S (P ) is one of the fundamental problems in computational geometry and graph theory [1,2,3,5,6,7,11]. Dillencourt [11] andÁbrego et al [1] considered the problem of matching points with disks.…”

Section: Previous Workmentioning

confidence: 99%

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Strong matching of points with geometric shapes

Biniaz

Maheshwari

Smid

2018

Computational Geometry

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Let P be a set of n points in general position in the plane. Given a convex geometric shape S, a geometric graph G S (P ) on P is defined to have an edge between two points if and only if there exists an empty homothet of S having the two points on its boundary. A matching in G S (P ) is said to be strong, if the homothests of S representing the edges of the matching, are pairwise disjoint, i.e., do not share any point in the plane. We consider the problem of computing a strong matching in G S (P ), where S is a diametral-disk, an equilateral-triangle, or a square. We present an algorithm which computes a strong matching in G S (P ); if S is a diametral-disk, then it computes a strong matching of size at least n−1 17 , and if S is an equilateral-triangle, then it computes a strong matching of size at least n−1 9 . If S can be a downward or an upward equilateral-triangle, we compute a strong matching of size at least n−1 4in G S (P ). When S is an axis-aligned square we compute a strong matching of size n−1 4in G S (P ), which improves the previous lower bound of n 5 .

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Section: Previous Workmentioning

confidence: 99%

“…Lemma 7 (Biniaz et al [7]). Let t 1 be a downward triangle which intersects a downward triangle t 2 through t 2 (s 1 ), and let a horizontal line intersects both t 1 and t 2 .…”

Section: Strong Matching In G (P )mentioning

confidence: 99%

Strong matching of points with geometric shapes

Biniaz

Maheshwari

Smid

2018

Computational Geometry

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“…See the following subsection for further background. Biniaz et al [12] proved that for any point set of size n, G (P ) has a strong matching of at least n−1 9 edges and G (P ) has a strong matching of at least (n − 1)/4 edges. We prove an upper bound on the size of a strong matching in Θ 6 -graphs by giving an example where the maximum strong matching in G (P ) has 2n/5 edges.…”

Section: Introductionmentioning

confidence: 99%

“…They also give lower bounds of (n − 1)/8 and n/5 , respectively. The lower bound for squares was improved to (n − 1)/4 by Biniaz et al [12] who also proved lower bounds of (n − 1)/9 for G and (n − 1)/4 for G .…”

Section: Introductionmentioning

confidence: 99%

Maximum Matchings and Minimum Blocking Sets in $$\varTheta _6$$ -Graphs

Biedl

Biniaz

Irvine

et al. 2019

Graph-Theoretic Concepts in Computer Science

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Θ6-Graphs graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is n/3, where n is the number of vertices of the graph. Babu et al. (2014) conjectured that any Θ6-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to (3n − 8)/7.We also relate the size of maximum matchings in Θ6-graphs to the minimum size of a blocking set. Every edge of a Θ6-graph on point set P corresponds to an empty triangle that contains the endpoints of the edge but no other point of P . A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least β(n)/2 where β(n) is the minimum, over all Θ6-graphs with n vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on β, allowing us to show that β(n) ≥ 3n/4 − 2.

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“…This implies that there exist triangle-DG graphs that do not contain Hamiltonian paths or cycles. Biniaz et al [21] showed that 7-DG △ contains a bottleneck Hamiltonian cycle and that there exist points sets where 5-DG △ does not contain a bottleneck Hamiltonian cycle. Ábrego et al [3] showed that the DG □ admits a Hamiltonian path, while Saumell [92] showed that the DG □ is not necessarily 1-tough, and Our contributions.…”

Section: Convex Shape Delaunay Graphs and Hamiltonicitymentioning

confidence: 99%

Generalized Delaunay Triangulations:Graph-Theoretic Properties and Algorithms

Vila¹,

Pilar²

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Higher-order triangular-distance Delaunay graphs: Graph-theoretical properties

Cited by 8 publications

References 20 publications

Strong matching of points with geometric shapes

Strong matching of points with geometric shapes

Maximum Matchings and Minimum Blocking Sets in $$\varTheta _6$$ -Graphs

Generalized Delaunay Triangulations:Graph-Theoretic Properties and Algorithms

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