Induced paths and cycles in factor graphs of split graphs

Let $S$ be a split graph with bipartition $(K,I)$ and let $\mathrm{\Phi }(S)$ be the factor graph associated with $S$, a multigraph on $I$ whose encodes the combinatorial information about 2-switch transformations in $S$. We study induced paths and cycles in $\mathrm{\Phi }(S)$ and show that they impose strong structural restrictions on the neighborhoods in $S$ of the corresponding vertices. In particular, induced paths generate chains of neighborhood inclusions which force a monotone behavior of the degrees (in $S$) of their vertices along the path. As a consequence, we prove that induced cycles in $\mathrm{\Phi }(S)$ have length $\le 4$. Finally, we show that in any induced path only the first or the last edge can be simple, which yields an upper bound for the diameter of $\mathrm{\Phi }(S)$ in terms of the 2-switch-degree of $S$.