Induced paths and cycles in factor graphs of split graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913008044736512 |
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| author | Schvöllner, Victor N. Pastine, Adrián |
| author_facet | Schvöllner, Victor N. Pastine, Adrián |
| contents | Let $S$ be a split graph with bipartition $(K,I)$ and let $Φ(S)$ 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 $Φ(S)$ 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 $Φ(S)$ 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 $Φ(S)$ in terms of the 2-switch-degree of $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14061 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Induced paths and cycles in factor graphs of split graphs Schvöllner, Victor N. Pastine, Adrián Combinatorics Let $S$ be a split graph with bipartition $(K,I)$ and let $Φ(S)$ 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 $Φ(S)$ 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 $Φ(S)$ 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 $Φ(S)$ in terms of the 2-switch-degree of $S$. |
| title | Induced paths and cycles in factor graphs of split graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.14061 |