Simple factor graphs associated with split graphs

TL;DR

This paper introduces the factor graph of split graphs, a simplified structure that captures 2-switch transformations, aiding in understanding their combinatorial properties and classifications.

Contribution

It presents a new, cleaner factor graph model for split graphs that improves upon previous representations and provides insights into their 2-switch dynamics.

Findings

Factor graph $\

provides a compact alternative to previous models.

When $\

Abstract

We introduce and study a loopless multigraph associated with a split graph $S$: the factor graph of $S$, denoted by $\Phi(S)$, which encodes the combinatorial information about 2-switch transformations over $S$. This construction provides a cleaner, compact and non-redundant alternative to the graph $A_4(S)$ by Barrus and West, for the particular case of split graphs. If $\Phi(S)$ is simple and connected, we obtain a precise description of the underlying structure of $S$, particularly when $\Phi(S)$ is complete, highlighting the usefulness of the factor graph for understanding 2-switch dynamics in balanced and indecomposable split graphs, as well as its 2-switch-degree classification.