Connectivity notions on compatible digraphs in equational classes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918393955745792 |
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| author | Gyenizse, Gergő Maróti, Miklós Zádori, László |
| author_facet | Gyenizse, Gergő Maróti, Miklós Zádori, László |
| contents | A digraph $\mathbb G$ is called weakly connected, strongly connected, and extremely connected if any two vertices of $\mathbb G$ are connected respectively by an oriented, a directed, and a symmetric path in $\mathbb G$. We investigate the algebraic properties of digraphs that force some of these connectivity notions to coincide.
We prove that for digraphs with a Hobby-McKenzie polymorphism, the strong and the extreme components coincide. Conversely, if the strong and the extreme components of any compatible digraph in an equational class of algebras coincide, then the class must have a Hobby-McKenzie term. As a consequence, we obtain that an equational class $\mathcal V$ is $n$-permutable for some $n$ if and only if the weak components of any compatible reflexive digraph in $\mathcal V$ are extremely connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16613 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Connectivity notions on compatible digraphs in equational classes Gyenizse, Gergő Maróti, Miklós Zádori, László Combinatorics Logic A digraph $\mathbb G$ is called weakly connected, strongly connected, and extremely connected if any two vertices of $\mathbb G$ are connected respectively by an oriented, a directed, and a symmetric path in $\mathbb G$. We investigate the algebraic properties of digraphs that force some of these connectivity notions to coincide. We prove that for digraphs with a Hobby-McKenzie polymorphism, the strong and the extreme components coincide. Conversely, if the strong and the extreme components of any compatible digraph in an equational class of algebras coincide, then the class must have a Hobby-McKenzie term. As a consequence, we obtain that an equational class $\mathcal V$ is $n$-permutable for some $n$ if and only if the weak components of any compatible reflexive digraph in $\mathcal V$ are extremely connected. |
| title | Connectivity notions on compatible digraphs in equational classes |
| topic | Combinatorics Logic |
| url | https://arxiv.org/abs/2603.16613 |