Towards a characterization of stretchable aligned graphs

TL;DR

This paper investigates the stretchability of pseudolines in planar drawings, demonstrating both limitations and conditions under which stretchability is guaranteed, thus advancing understanding of aligned graph representations.

Contribution

It proves that not all two-pseudoline instances are stretchable and identifies conditions for stretchability in arrangements with intersecting pseudolines.

Findings

Not all two-pseudoline instances are stretchable.

Certain intersection patterns guarantee stretchability.

Aligned graphs with intersection-free arrangements are always stretchable.

Abstract

We consider the problem of stretching pseudolines in a planar straight-line drawing to straight lines while preserving the straightness and the combinatorial embedding of the drawing. We answer open questions by Mchedlidze et al. by showing that not all instances with two pseudolines are stretchable. On the positive side, for $k\geq 2$ pseudolines intersecting in a single point, we prove that in case that some edge-pseudoline intersection-patterns are forbidden, all instances are stretchable. For intersection-free pseudoline arrangements we show that every aligned graph has an aligned drawing. This considerably reduces the gap between stretchable and non-stretchable instances.