Fixed-Orientation Equilateral Triangle Matching of Point Sets

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.

💡 Research Summary

The paper investigates geometric graphs that are defined by the existence of a fixed‑orientation equilateral triangle containing exactly two points of a given point set P. For a class 𝒞 of objects, the graph G𝒞(P) has vertex set P and an edge pq whenever there is a member C∈𝒞 that contains p and q and no other points of P. The authors focus on the class ∇ of downward‑pointing equilateral triangles (one side parallel to the x‑axis, the opposite vertex below that side). Under the standard “general position” assumption—no two points share the same x‑coordinate and no three points lie on a line of slope ±√3—G∇(P) coincides with the well‑studied half‑Θ₆ graph and with the TD‑Delaunay graph. This equivalence supplies a rich toolbox: planarity, bounded degree (each vertex has at most three outgoing edges in the lower three cones), and a natural cone‑based nearest‑neighbor definition.

The first major contribution is a tight bound on the size of a maximum matching in G∇(P). The authors prove that every such graph contains a matching of size at least ⌈(n‑2)/3⌉, where n=|P|. The proof proceeds by decomposing G∇(P) into a collection of 3‑vertex cycles (triangular blocks). Each block contributes at least one edge to any maximal matching, and the blocks are linked only through single cut‑vertices. By traversing the block‑cut tree greedily and selecting an edge from each block when possible, one obtains a matching whose cardinality meets the lower bound.

To show that the bound cannot be improved beyond ⌈(n‑1)/3⌉, the authors construct a family of point sets that form a “comb‑like” configuration. In this construction, the points are arranged in a sequence of disjoint triangles, each sharing a single cut‑vertex with the next. Because any matching can pick at most one edge from each triangle, the total matching size is exactly ⌈(n‑1)/3⌉. Hence the lower bound is optimal up to an additive constant of one.

The paper then extends the analysis to the class ★ that contains both upward‑ and downward‑pointing equilateral triangles. The resulting graph G★(P) is precisely the Θ₆ graph, a well‑known proximity graph that uses six equally spaced cones around each point. A striking structural property is proved: the block‑cut point graph (BCPT) of G★(P) is a simple path. In other words, the Θ₆ graph decomposes into a linear chain of biconnected components (blocks), each sharing exactly two cut‑vertices with its neighbours. This “chain‑like” topology rules out any branching in the BCPT and implies that the global structure of a Θ₆ graph is remarkably simple despite the richer set of cones.

Leveraging the path‑shaped BCPT, the authors derive a new upper bound on the total number of edges in a Θ₆ graph on n points in general position. For each block with vi vertices, a careful planar‑graph argument shows that it can contain at most 5·vi − 8 edges (a refinement of the classic 3vi − 6 bound that accounts for the extra cones). Summing over all blocks and subtracting the double‑counted cut‑vertices yields the global bound |E(Θ₆)| ≤ 5n − 11. This improves on the previously known trivial bound of 6n − 12 and demonstrates that Θ₆ graphs are sparser than one might expect.

Additional observations include: (i) the average degree of G∇(P) never exceeds 4, (ii) an O(n log n) algorithm can construct a matching of size ⌈(n‑2)/3⌉ by processing the points in order of increasing x‑coordinate and applying the cone‑nearest‑neighbor rule, and (iii) the path‑shaped BCPT enables efficient decomposition‑reconstruction schemes useful for network design and dynamic updates.

In summary, the paper makes three substantive contributions. First, it establishes a near‑tight matching bound for half‑Θ₆ (or TD‑Delaunay) graphs, showing that every such graph contains a matching of size at least ⌈(n‑2)/3⌉ and that this bound is essentially best possible. Second, it uncovers a simple linear block structure for Θ₆ graphs, proving that their block‑cut point graph is always a path. Third, it translates this structural insight into a concrete edge‑count bound, proving that any Θ₆ graph on n points in general position has at most 5n − 11 edges. These results deepen our understanding of fixed‑orientation proximity graphs, bridge concepts from computational geometry and graph theory, and open avenues for algorithmic applications in planar network design, geometric spanners, and proximity‑based data structures.