Simple factor graphs associated with split graphs

TL;DR

This work introduces the factor graph $Φ(S)$ for split graphs, encoding 2-switch activity between independent vertices via edge multiplicities $σ_{uv}(S)$ and offering a more succinct alternative to the $A_4(S)$ construction. It establishes fundamental links between $Φ(S)$ and Barrus–West’s $A_4(S)$, showing that $Φ(S)$-connectedness implies indecomposability (and primeness for active graphs), while $A_4(S)$-connectedness implies $Φ(S)$-connectedness; conversely, the latter implication requires additional hypotheses. A key result is the closed form $σ_{uv}(S)=(d_u-\,η_{uv})(d_v-\,η_{uv})$, tying multiplicities to neighborhoods and enabling a notion of homogeneous split graphs. The paper then analyzes the case when $Φ(S)$ is simple and complete, deriving an intersecting-family structure for the neighborhoods and proving a precise, Tyshkevich-based classification: active, simple-complete cases correspond to $S≅G$ or $S≅ar{G^{}}$, with a constructive formulation via a degree-1 base graph $R$. Finally, for general simple connected $Φ(S)$, the authors show that removing universal vertices does not affect $Φ$, establish degree/ clique-number relations, and provide a complete characterization of active split graphs with simple connected $Φ$ as those isomorphic to $R$ or $ar{R^{}}$.

Abstract

We introduce and study a loopless multigraph associated with a split graph $S$: the factor graph of $S$, denoted by $Φ(S)$, 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 $Φ(S)$ is simple and connected, we obtain a precise description of the underlying structure of $S$, particularly when $Φ(S)$ 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.

Figures (4)

• Figure 1: A split graph $S$ and its associated $A_4$ and $\Phi$.
• Figure 2: $\deg(S)=\sigma_{ab}+\sigma_{ac}+\sigma_{bc}=4+6+0=10$.
• Figure 3: $N_{\Phi}(1)=N_{\Phi}(2)$ but $N_1\neq N_2$.
• Figure 4: $N_2\subset N_1$ but $N_{\Phi}(2)\nsubseteq N_{\Phi}(1)$.

Theorems & Definitions (45)

• Theorem 1.1: Tyshkevich, tyshkevich2000decomposition
• Theorem 1.2: Barrus, West, barrus.west.A4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof