arXiv subjects

Victor Nicolas Schvöllner

Publications and source records attributed to Victor Nicolas Schvöllner.

3 recordsLinked to original sources

arXiv2026

Simple factor graphs associated with split graphs

We introduce and study a loopless multigraph associated with a split graph $(S,K,I)$: its factor graph $Φ(S,K,I)$, 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.

math.CO↗

arXiv2025

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

math.CO↗

arXiv2020

2-switch: transition and stability on graphs and forests

Given any two forests with the same degree sequence, we show in an algorithmic way that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests. We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequence. Then, we apply these results to conclude that the studied parameters have the interval property on those families.

math.CO↗