Given a point set P and a class of geometric objects, 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 containing both p and q but no other points from P. We study 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- graphs and TD-Delaunay graphs.
The main result in our paper is that for point sets P in general position, always contains a matching of size at least and this bound is tight. We also give some structural properties of 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 is simply a path. Through the equivalence of graphs with graphs, we also derive that any graph can have at most edges, for point sets in general position.