  Given a point set $P$ and a class $\mathcal{C}$ of geometric objects,
$G_\mathcal{C}(P)$ 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 $C \in
\mathcal{C}$ containing both $p$ and $q$ but no other points from $P$. We study
$G_{\bigtriangledown}(P)$ graphs where $\bigtriangledown$ 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-$\Theta_6$ graphs and TD-Delaunay graphs.
  The main result in our paper is that for point sets $P$ in general position,
$G_{\bigtriangledown}(P)$ always contains a matching of size at least
$\lceil\frac{n-2}{3}\rceil$ and this bound cannot be improved above
$\lceil\frac{n-1}{3}\rceil$.
  We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where
$\davidsstar$ 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 $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of
$G_{\davidsstar}(P)$ graphs with $\Theta_6$ graphs, we also derive that any
$\Theta_6$ graph can have at most $5n-11$ edges, for point sets in general
position.

Computational Geometry;Discrete Mathematics