Connected Matchings

TL;DR

This work investigates the largest guaranteed connected straight-line matchings in planar point sets, introducing f(n) for uncolored sets and g(n,c) for balanced c-colorings. It develops a linear-time separatrix framework based on separating paths and triangle-splitting lemmas, enabling a constructive lower bound f(n) ≥ (5n+1)/27 computable in O(n log n), while establishing a general upper bound f(n) ≤ ⌈(n-1)/3⌉. For colored settings, it provides two polychromatic separating-path results yielding g(n,c) ≳ (c−3)n/(6c) for large c and g(n,2) ≥ n/18−1/3, with linear-time algorithms to realize these bounds. The paper also presents depth-based and “deep-point” arguments to tighten lower bounds and discusses extensions to stronger connectivity and related substructures. Overall, it advances the understanding of how many edges can form a connected crossing network on a given point set, and provides efficient algorithms to compute such matchings.

Abstract

We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, and such a matching can be computed in $O(n \log n)$ time. As an upper bound, we show that for some planar point sets in general position the largest matching whose segments form a connected set has $\lceil \frac{n-1}{3}\rceil$ edges. We also consider a colored version, where each edge of the matching should connect points with different colors.

Figures (14)

• Figure 1: Proof of \ref{['lem:CH']}. The pair $(a,b)$ defining the blue line is a feasible solution to the linear program.
• Figure 2: Proof of \ref{['lem:maximal']}. The definition of the transformation $\varphi=(\varphi_1,\varphi_2)$.
• Figure 3: Statement in \ref{['thm:split-triangle']}.
• Figure 4: Left: rotating $r_1$ until we pass over $m-w_1-1$ points. Any point not scanned by $r_1$ defines with $p_0$ and $p_2$ a triangle with at most $w_1$ points. Right: the part of the triangle that is not shadowed contains at least $\ell-2m+3$ points.
• Figure 5: Left: $5$-separating path of length $1$. Right: $7$-separating path of length $2$.

Theorems & Definitions (24)

• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Corollary 6
• proof
• Theorem 7
• proof