Order-preserving 1-string representations of planar graphs

TL;DR

This paper investigates order-preserving 1-string representations of planar graphs, demonstrating their existence for certain subclasses like outer-planar graphs, but not for all planar graphs, including some 3-trees.

Contribution

It establishes the non-existence of order-preserving 1-string representations for all planar graphs and provides constructions for specific subclasses such as outer-planar graphs.

Findings

Order-preserving 1-string representations do not exist for all planar graphs.

Such representations exist for some subclasses like outer-planar graphs.

Outer-string representations can be achieved with all string ends on the outside.

Abstract

This paper considers 1-string representations of planar graphs that are order-preserving in the sense that the order of crossings along the curve representing vertex $v$ is the same as the order of edges in the clockwise order around $v$ in the planar embedding. We show that this does not exist for all planar graphs (not even for all planar 3-trees), but show existence for some subclasses of planar partial 3-trees. In particular, for outer-planar graphs it can be order-preserving and outer-string in the sense that all ends of strings are on the outside of the representation.