Fixed-Orientation Equilateral Triangle Matching of Point Sets

TL;DR

This paper studies geometric graphs formed by connecting points with downward equilateral triangles, establishing bounds on matchings and structural properties, and relating these graphs to well-known geometric graph classes.

Contribution

It proves lower bounds on matchings in downward equilateral triangle graphs and explores structural properties of combined upward and downward triangle graphs, linking them to $ heta_6$ graphs.

Findings

Matching size at least $rac{n-2}{3}$ for downward triangle graphs.

Block cut point graph of combined triangle graphs is a path.

Maximum edges in $ heta_6$ graphs is $5n-11$.

Abstract

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.