Graphs of scramble number two

TL;DR

This paper classifies graphs with scramble number at most two using forbidden topological minors and shows that such a finite classification does not exist for higher scramble numbers.

Contribution

It provides a complete classification of graphs with scramble number at most two and proves the non-existence of a finite forbidden minor characterization for higher scramble numbers.

Findings

Graphs with scramble number ≤ 2 are characterized by a finite set of forbidden minors.

No finite forbidden minor characterization exists for graphs with scramble number > 2.

The scramble number bounds are related to gonality and treewidth.

Abstract

The scramble number of a graph provides a lower bound for gonality and an upper bound for treewidth, making it a graph invariant of interest. In this paper we study graphs of scramble number at most two, and give a classification of all such graphs with a finite list of forbidden topological minors. We then prove that there exists no finite list of forbidden topological minors to characterize graphs of any fixed scramble number greater than two.