Strong matching of points with geometric shapes

Abstract

Let P be a set of n points in general position in the plane. Given a convex geometric shape S, a geometric graph $G_S(P)$ on P is defined to have an edge between two points if and only if there exists a homothet of S having the two points on its boundary and whose interior is empty of points of P. A matching in $G_S(P)$ is said to be strong, if the homothets of S representing the edges of the matching are pairwise disjoint, i.e., they do not share any point in the plane. We consider the problem of computing a strong matching in $G_S(P)$, where S is a diametral disk, an equilateral triangle, or a square. We present an algorithm that computes a strong matching in $G_S(P)$; if S is a diametral-disk, then it computes a strong matching of size at least $\lceil \frac{n-1}{17}\rceil$, and if S is an equilateral-triangle, then it computes a strong matching of size at least $\lceil \frac{n-1}{9}\rceil$. If S can be a downward or an upward equilateral-triangle, we compute a strong matching of size at least $\lceil \frac{n-1}{4}\rceil$ in $G_S(P)$. When S is an axis-aligned square, we compute a strong matching of size at least $\lceil \frac{n-1}{4}\rceil$ in $G_S(P)$, that improves the previous lower bound of $\lceil \frac{n}{5}\rceil$.