Minimal crossing diagrams of spatial graphs

TL;DR

This paper proves that certain classes of spatial graphs have minimal crossing diagrams and provides examples addressing open questions about minimal crossing representations of rigid vertex graphs.

Contribution

It establishes minimal crossing number results for 1-vertex spatial graphs and diagrams derived from planar graphs, and responds to a question on rigid vertex graphs.

Findings

All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.

Diagrams obtained from planar graphs by replacing vertices and edges with minimal crossing diagrams also have minimal crossing number.

Provided an example answering a question about minimal crossing diagrams of rigid vertex graphs.

Abstract

We prove that all $1$-vertex spatial graphs with adequate diagrams have minimal crossing number, and that spatial graph diagrams obtained by replacing vertices and edges of a planar embedded graph by minimal crossing link or spatial graph diagrams have minimal crossing number. Finally, we give an example in answer to a question of Adams et al. about minimal crossing diagrams of rigid vertex graphs.