Algorithms and Bounds for Drawing Non-planar Graphs with Crossing-free Subgraphs

TL;DR

This paper explores conditions under which non-planar graphs can be drawn with a planar subgraph crossing-free, considering straight-line and bent-edge drawings, and provides results on existence and trade-offs.

Contribution

It introduces the problem of drawing non-planar graphs with a crossing-free planar subgraph and analyzes solutions with and without edge bends.

Findings

Existence of straight-line drawings with crossing-free planar subgraphs for certain classes.

Trade-offs between the number of bends and drawing area to achieve crossing-free subgraphs.

Positive and negative results depending on the type of spanning subgraph.

Abstract

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing {\Gamma} of G in the plane such that the edges of S are not crossed in {\Gamma} by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.