Fixed-orientation equilateral triangle matching of point sets

Abstract

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.

Matchings in higher-order Gabriel graphs

“…The bound in Theorem 7 is tight, as can be seen from the graph in Fig. 9, for which the maximum matching has size n−1 4 . Actually this is a Gabriel graph of maximum degree four which is a tree.…”

“…We also showed that n−1 2 points are necessary and n − 1 points are sufficient to block 0-TD. In [4] it is shown that 0-TD has a matching of size n−1 3 .…”

“…4 and let x be a point in X . For i = 1, 2, 3, 4, let c i be the center of D i , let C i be the boundary of D i , and let (a i , b i ) be the edge in T whichcorresponds to D i .…”

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

“…Lemma 1 (Babu et al [8]). Let P be a set of points in the plane, and let p and q be any two points in P .…”

“…In fact, they might fall short of being triangulations. As discussed by Drysdale [22] and Chew [18] (see also [7]), they are plane graphs that consist of a "support hull" which need not be convex, and a complete triangulation of the interior (an explicit proof can be found in [8]). This anomaly is often remedied by surrounding the point set with a large bounding triangle.…”