# Fixed-Orientation Equilateral Triangle Matching of Point Sets

Ahmad Biniaz

Given a set *P* of *n* points 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 upward equilateral triangles. For point sets in general position, these graphs have
been shown to be equivalent to half-Θ_{6} graphs and TD-Delaunay graphs.
We show that *G*_{△}(P) contains a matching of size at least ⌈(*n*-1)/3⌉ and this
bound is tight.
We also give some structural properties of Θ_{6} graphs, which contains both upward and downward equilateral triangles.
We show that the block cut point
graph of Θ_{6} is simply a path. We also show that any Θ_{6} can have at
most 5*n* - 11 edges. Finally we prove that Θ_{6} has an strong matching of size at least ⌈*n*/5⌉ where triangles related to the edges of the matching do not overlap.

This is joint work with Jasine Babu, Anil Maheshwari, and Michiel Smid.