Elsevier

Theoretical Computer Science

Volume 555, 23 October 2014, Pages 55-70
Theoretical Computer Science

Fixed-orientation equilateral triangle matching of point sets

https://doi.org/10.1016/j.tcs.2013.11.031Get rights and content
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Abstract

Given a point set P and a class C of geometric objects, GC(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 CC containing both p and q but no other points from P. We study G(P) graphs where ▽ is the class of downward equilateral triangles (i.e., 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-Θ6 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 least |P|13 and 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 Θ6 graphs, we also derive that any Θ6 graph can have at most 5n11 edges, for point sets in general position.

Keywords

Geometric graphs
Delaunay graphs
Half-Θ6 graphs
Matching

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