Fixed-Orientation Equilateral Triangle Matching of Point Sets

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 C ∈ C containing both p and q but no other points from P . We study G5(P ) graphs where 5 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-Θ6 graphs and TD-Delaunay graphs. The main result in our paper is that for point sets P in general position, G5(P ) always contains a matching of size at least ⌈ n−2 3 ⌉ and this bound cannot be improved above ⌈ n−1 3 ⌉ . We also give some structural properties of GC(P ) graphs, where C 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 GC(P ) is simply a path. Through the equivalence of GC(P ) graphs with Θ6 graphs, we also derive that any Θ6 graph can have at most 5n− 11 edges, for point sets in general position.