# Packing Short Plane Spanning Graphs in Complete Geometric Graphs

## TL;DR

This paper develops methods to create multiple disjoint, plane, and near-minimal spanning layers in complete geometric graphs, with both centralized and distributed algorithms suitable for sensor networks.

## Contribution

It introduces an almost optimal centralized algorithm and a constant factor approximation for a distributed model to generate layered spanning graphs.

## Key findings

- Centralized approach efficiently extracts two plane spanning graphs.
- Distributed algorithm achieves constant factor approximation with local information.
- Both methods produce plane layers suitable for sensor network applications.

## Abstract

Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is very long (when compared to the minimum length needed to obtain a spanning graph). We consider two different approaches: first we show an almost optimal centralized approach to extract two graphs. Then we show a constant factor approximation for a distributed model in which each point can compute its adjacencies using only local information. In both cases the obtained layers are plane