Abstract:We introduce and study a loopless multigraph associated with a split graph $(S,K,I)$: its factor graph $\Phi(S,K,I)$, whose vertex set is $I$ and whose edges, counted with multiplicity, record the 2-switches acting on $S$. In contrast with the $A_4$-structure $\mathcal{A}_4(S)$ introduced by Barrus and West, the factor graph is supported on the independent set alone; it is therefore more compact and, although it retains less information, it still determines the 2-switch-degree of $S$. We show that an active split graph is indecomposable if and only if its factor graph is connected, an analogue for split graphs of the Barrus--West characterization of indecomposable graphs. Finally, we describe completely the active split graphs whose factor graph is simple and connected.
From: Victor Nicolas Schvöllner [view email]
[v1]
Tue, 30 Dec 2025 14:26:33 UTC (76 KB)
[v2]
Thu, 3 Sep 2026 07:40:42 UTC (26 KB)
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