Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

TL;DR

This paper develops algorithms to construct planar geometric graphs with bounded edge length and specific connectivity properties, achieving near-optimal solutions and establishing bounds on what is possible under various conditions.

Contribution

It introduces efficient algorithms for constructing 2-edge connected planar graphs with bounded edge length, and proves bounds on the approximation factor achievable.

Findings

Constructed a minimum degree 2 planar graph with max edge length ≤ 2 times optimal in O(n log n)

Built a 2-edge connected planar graph with max edge length ≤ √5 times optimal assuming P forms a connected Unit Disk Graph

Proved that a factor of 2 times optimal is always sufficient for 2-edge connected graphs in certain conditions

Abstract

Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set $P$, respectively. First, we construct in $O(n\log{n})$ time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in $O(n\log n)$ time a 2-edge connected geometric planar graph spanning $P$ with max edge length bounded by $\sqrt{5}$ times the optimal, assuming that the set $P$ forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in $O(n^2)$ time. We also show that for $k \in O(\sqrt{n})$, there exists a set $P$ of $n$ points in the plane such that even though the Unit Disk Graph spanning $P$ is $k$-vertex connected, there is no 2-edge connected geometric planar graph spanning $P$ even if the length of its edges is allowed to be up to 17/16.