Induced paths and cycles in factor graphs of split graphs

TL;DR

This paper investigates the structure of factor graphs derived from split graphs, revealing restrictions on induced paths and cycles that influence neighborhood configurations and graph diameter.

Contribution

It establishes bounds on induced cycle lengths and the diameter of factor graphs based on 2-switch degrees, providing new structural insights.

Findings

Induced cycles in the factor graph have length at most 4.

Induced paths generate neighborhood inclusion chains leading to degree monotonicity.

The diameter of the factor graph is bounded by the 2-switch-degree of the split graph.

Abstract

Let $S$ be a split graph with bipartition $(K,I)$ and let $\Phi(S)$ 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 $\Phi(S)$ 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 $\Phi(S)$ have length $\leq 4$. 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 $\Phi(S)$ in terms of the 2-switch-degree of $S$.