Unknottability of spatial graphs by region crossing changes

TL;DR

This paper characterizes when spatial graphs derived from planar graphs can be simplified to an unknot using region crossing changes, revealing a complete classification based on Eulerian properties.

Contribution

It provides a necessary and sufficient condition for unknottability of spatial graphs via region crossing changes, linking graph Eulerian properties to topological simplification.

Findings

Spatial graphs from planar graphs are unknottable iff they are non-Eulerian or Eulerian and proper.

The paper establishes a complete classification of unknottability based on Eulerian conditions.

Provides a topological criterion connecting graph properties to spatial graph simplification.

Abstract

A region crossing change is a local transformation on spatial graph diagrams switching the over/under relations at all the crossings on the boundary of a region. In this paper, we show that a spatial graph of a planar graph is unknottable by region crossing changes if and only if the spatial graph is non-Eulerian or is Eulerian and proper.