On plane subgraphs of complete topological drawings

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 ⌈3 n /2⌉ 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.