Essential Constraints of Edge-Constrained Proximity Graphs

TL;DR

This paper introduces efficient algorithms for constructing minimal constraint sets in proximity graphs, ensuring specific subgraphs are contained within edge-constrained minimum spanning trees and related structures, with applications to various geometric graphs.

Contribution

It presents an $O(n ext{ log } n)$ algorithm for finding minimal constraint sets in edge-constrained MSTs and generalizes to other proximity graphs like Gabriel and Delaunay graphs, with an $O(n)$ solution given a constrained Delaunay triangulation.

Findings

Efficient $O(n ext{ log } n)$ algorithm for edge-constrained MST constraints.

Generalization to proximity graphs like Gabriel and $eta$-skeletons.

Linear time algorithm when the constrained Delaunay triangulation is provided.

Abstract

Given a plane forest $F = (V, E)$ of $|V| = n$ points, we find the minimum set $S \subseteq E$ of edges such that the edge-constrained minimum spanning tree over the set $V$ of vertices and the set $S$ of constraints contains $F$. We present an $O(n \log n )$-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, $\beta$-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set $S\subseteq E$ of edges of a given plane graph $I=(V,E)$ such that $I \subseteq CG_\beta(V, S)$ for $1 \leq \beta \leq 2$, where $CG_\beta(V, S)$ is the constraint $\beta$-skeleton over the set $V$ of vertices and the set $S$ of constraints. The running time of our algorithm is $O(n)$, provided that the constrained Delaunay triangulation of $I$ is given.