2-switch-degree classification of split graphs

TL;DR

This paper investigates the structural properties of 2-switch-degree in split graphs, introduces a novel factor graph approach for classification, and establishes a surprising link between graph theory and number theory through the $\Delta$-property.

Contribution

It presents a new classification method for indecomposable split graphs using factor graphs and fully classifies those with degrees up to 4, revealing a novel connection to number theory.

Findings

Classified indecomposable split graphs of degrees 1 to 4.

Introduced the factor graph $\Phi(S)$ to encode 2-switch-degree information.

Discovered the $\Delta$-property linking graph theory and number theory.

Abstract

The 2-switch-degree of $G$ is the number of distinct 2-switches acting on a graph $G$. In this work we study structural properties of the 2-switch-degree, with a focus on split graphs. Our approach is motivated by the Tyshkevich decomposition, which uniquely expresses any graph as a composition $G_r \circ \ldots \circ G_1$ of indecomposable graphs, where $G_2, \ldots, G_r$ are split. Our key tool is the factor graph $\Phi(S)$, a multigraph associated with a split graph $S$ that encodes 2-switch-degree information via edge multiplicities between independet vertices of $S$. By leveraging $\Phi(S)$, we reduce the problem of classifying indecomposable split graphs to enumerating and analyzing unlabeled connected multigraphs of fixed size. Using this method, we fully classify indecomposable split graphs of degrees 1, 2, 3, and 4. Further, we introduce and investigate the $\Delta$-property, a surprising connection between Graph Theory and Number Theory that arises from $n$-simple triangles (3-cycles with uniform edge multiplicity $n$) of the factor graph.