On Plane Subgraphs of Complete Topological Drawings

TL;DR

This paper investigates properties of maximal plane subgraphs within simple topological drawings of complete graphs, establishing structural characteristics, computational complexity results, and algorithms for augmenting such subgraphs.

Contribution

It characterizes the structure of maximal plane subgraphs, proves NP-completeness of finding maximum plane subgraphs, and provides efficient algorithms for augmentation.

Findings

Maximal plane subgraphs are 2-connected and essentially 3-edge-connected.

Maximum plane subgraph problem is NP-complete.

A maximum plane augmentation can be found in O(n^3) time.

Abstract

Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a common endpoint or at a proper crossing. In this paper we study properties of maximal plane subgraphs of simple drawings $D_n$ of the complete graph $K_n$ on $n$ vertices. Our main structural result is that maximal plane subgraphs are 2-connected and what we call essentially 3-edge-connected. Besides, any maximal plane subgraph contains at least $\lceil 3n/2 \rceil$ edges. We also address the problem of obtaining a plane subgraph of $D_n$ with the maximum number of edges, proving that this problem is NP-complete. However, given a plane spanning connected subgraph of $D_n$, a maximum plane augmentation of this subgraph can be found in $O(n^3)$ time. As a side result, we also show that the problem of finding a largest compatible plane straight-line graph of two labeled point sets is NP-complete.