Toughness and Delaunay triangulations

Dillencourt, Michael B.

doi:10.1145/41958.41978

Cited by 29 publications

(38 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the proof of the next result we will use the following result of Dillencourt [11]: Let Γ be a Delaunay drawing with possibly degenerate vertex set V . Add, if necessary, edges to Γ to obtain a triangulation of V .…”

Section: Approximate Delaunay Drawingsmentioning

confidence: 99%

See 1 more Smart Citation

Approximate Proximity Drawings

et al. 2012

View full text Add to dashboard Cite

We introduce and study a generalization of the well-known region of influence proximity drawings, called (ε 1 , ε 2 )-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε 1 ≥ 0 and ε 2 ≥ 0, an (ε 1 , ε 2 )-proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u, v, their proximity region "shrunk" by the multiplicative factor 1 1+ε 1 does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u, v, their proximity region "expanded" by the factor (1 + ε 2 ) contains some vertices of Γ other than u and v. In particular, the locations of the vertices in such a drawing do not always completely determine which edges must be present/absent, giving us some freedom of choice. We show that this generalization significantly enlarges the family of representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of ε 1 and ε 2 . We also study the extremal case of (0, ε 2 )-proximity drawings, which generalize the well-known weak proximity drawing paradigm.

show abstract

Section: Approximate Delaunay Drawingsmentioning

confidence: 99%

“…Note that this immediately implies that the graph with the planar embedding of Fig. 4 does not admit an embedding preserving Delaunay drawing [11] since removing vertices a, b, c and d yields three connected components.…”

Section: Approximate Delaunay Drawingsmentioning

confidence: 99%

Approximate Proximity Drawings

et al. 2012

View full text Add to dashboard Cite

show abstract

“…In our study, we generally use the edges of an approximate Delaunay triangulation of the set of points. This edge set has the nice property that it is not too dense, but still captures well the structure of the point set and it contains a perfect matching (see Akl [2] and Dillencourt [21] ).…”

Section: Price and Repairmentioning

confidence: 99%

Computing Minimum-Weight Perfect Matchings

Cook

Rohe

1999

INFORMS Journal on Computing

205

175

View full text Add to dashboard Cite

We make several observations on the implementation of Edmonds' blossom algorithm for solving minimum-weight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dual-change ⑀ for each tree. As a benchmark of the algorithm's performance, solving a 100,000-node geometric instance on a 200 Mhz Pentium-Pro computer takes approximately 3 minutes.

show abstract

“…Given a point set P and a class C of geometric objects, the maximum C -matching problem is to compute a subclass C ′ of C of maximum cardinality such that no point from P belongs to more than one element of C ′ and for each C ∈ C ′ , there are exactly two points from P which lie inside C. Dillencourt [4] proved that every point set admits a perfect circle-matching.Ábrego et al [5] studied the isothetic square matching problem. Bereg et al concentrated on matching points using axis-aligned squares and rectangles [6].…”

Section: Introductionmentioning

confidence: 99%

Fixed-orientation equilateral triangle matching of point sets

Babu

Biniaz

Maheshwari

et al. 2014

Theoretical Computer Science

View full text Add to dashboard Cite

Given a point set P and a class C of geometric objects, G C (P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P. We study G ▽ (P) graphs where ▽ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ▽ (P) always contains a matching of size at leastand this bound is tight. We also give some structural properties of G (P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G (P) is simply a path. Through the equivalence of G (P) graphs with Θ 6 graphs, we also derive that any Θ 6 graph can have at most 5n − 11 edges, for point sets in general position.

show abstract

Toughness and Delaunay triangulations

Cited by 29 publications

References 23 publications

Approximate Proximity Drawings

Approximate Proximity Drawings

Computing Minimum-Weight Perfect Matchings

Fixed-orientation equilateral triangle matching of point sets

Contact Info

Product

Resources

About