Fixed-Orientation Equilateral Triangle Matching of Point Sets

Abstract

Given a point set P 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 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-Θ 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ▽ (P) always contains a matching of size at least |P|−1 3 and this bound is tight. We also give some structural properties of G (P) graphs, where 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 (P) is simply a path. Through the equivalence of G (P) graphs with Θ 6 graphs, we also derive that any Θ 6 graph can have at most 5n − 11 edges, for point sets in general position.

Strong matching of points with geometric shapes

“…The problem of computing a maximum matching in G S (P ) is one of the fundamental problems in computational geometry and graph theory [1,2,3,5,6,7,11]. Dillencourt [11] andÁbrego et al [1] considered the problem of matching points with disks.…”

“…Otherwise, by induction, we get a matching of size (m 1 + 1) + m 2 + (m 3 4 . In all cases we get a matching of size m + 1 in S.…”

Higher-order triangular-distance Delaunay graphs: Graph-theoretical properties

“…For a set P of points in the plane, the TD-Delaunay graph, i.e., 0-TD, is not necessarily a triangulation [12], but it is connected and internally triangulated [4]. As shown in Figure 1(a), the outer face may not be convex and hence 0-TD is not necessarily biconnected.…”

“…In [4] the authors proved a tight lower bound of n−1 3 on the size of a maximum matching in 0-TD. In this section we prove that 1-TD has a matching of size 2(n−1) 5 and 2-TD has a perfect matching when P has an even number of points.…”

“…In [4], the authors proved a tight lower bound of n−1 3 on the size of a maximum matching in a TD-Delaunay graph. In this paper we study higher-order TD-Delaunay graphs.…”