2-switch-degree classification of split graphs

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 $G_r\circ \dots \circ G_1$ of indecomposable graphs, where $G_2,\dots ,G_r$ are split. Our key tool is the factor graph $\mathrm{\Phi }(S)$, a multigraph associated with a split graph $S$ that encodes 2-switch-degree information via edge multiplicities between independet vertices of $S$. By leveraging $\mathrm{\Phi }(S)$, 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 $\mathrm{\Delta }$-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.