Edge Partitions of Complete Geometric Graphs (Part 2)

TL;DR

This paper explores partitioning complete geometric graphs into subgraphs with relaxed planarity constraints, providing initial bounds for such partitions into k-planar and k-quasi-planar subgraphs.

Contribution

It extends previous work by establishing first bounds on partitioning into beyond planar subgraphs like k-planar and k-quasi-planar graphs.

Findings

First bounds on the number of subgraphs needed for k-planar partitions.

Extension of partitioning concepts to beyond planar subgraphs.

Advancement in understanding geometric graph partitions.

Abstract

Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into $k$-planar and $k$-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.