On the inclusion ideal graph of a poset
Let \((P, \leq)\) be an atomic partially ordered set (poset, briefly) with a minimum element \(0\) and \(\mathcal{I}(P)\) the set of nontrivial ideals of \( P \). The inclusion ideal graph of \(P\), denoted by \(\Omega(P)\), is an undirected and simple graph with the vertex set \(\mathcal{I}(P)\) an...
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| Datum: | 2019 |
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| Sprache: | English |
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Lugansk National Taras Shevchenko University
2019
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/261 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-article-2612019-07-14T19:54:06Z On the inclusion ideal graph of a poset Jahanbakhsh, N. Nikandish, R. Nikmehr, M. J. poset, inclusion ideal graph, diameter, girth, connectivity 06A07; 05C25 Let \((P, \leq)\) be an atomic partially ordered set (poset, briefly) with a minimum element \(0\) and \(\mathcal{I}(P)\) the set of nontrivial ideals of \( P \). The inclusion ideal graph of \(P\), denoted by \(\Omega(P)\), is an undirected and simple graph with the vertex set \(\mathcal{I}(P)\) and two distinct vertices \(I, J \in \mathcal{I}(P) \) are adjacent in \(\Omega(P)\) if and only if \( I \subset J \) or \( J \subset I \). We study some connections between the graph theoretic properties of this graph and some algebraic properties of a poset. We prove that \(\Omega(P)\) is not connected if and only if \( P = \{0, a_1, a_2 \}\), where \(a_1, a_2\) are two atoms. Moreover, it is shown that if \( \Omega(P) \) is connected, then \( \operatorname{diam}(\Omega(P))\leq 3 \). Also, we show that if \( \Omega(P) \) contains a cycle, then \( \operatorname{girth}(\Omega(P)) \in \{3,6\}\). Furthermore, all posets based on their diameters and girths of inclusion ideal graphs are characterized. Among other results, all posets whose inclusion ideal graphs are path, cycle and star are characterized. Lugansk National Taras Shevchenko University 2019-07-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/261 Algebra and Discrete Mathematics; Vol 27, No 2 (2019) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/261/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/261/542 Copyright (c) 2019 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2019-07-14T19:54:06Z |
| collection |
OJS |
| language |
English |
| topic |
poset inclusion ideal graph diameter girth connectivity 06A07 05C25 |
| spellingShingle |
poset inclusion ideal graph diameter girth connectivity 06A07 05C25 Jahanbakhsh, N. Nikandish, R. Nikmehr, M. J. On the inclusion ideal graph of a poset |
| topic_facet |
poset inclusion ideal graph diameter girth connectivity 06A07 05C25 |
| format |
Article |
| author |
Jahanbakhsh, N. Nikandish, R. Nikmehr, M. J. |
| author_facet |
Jahanbakhsh, N. Nikandish, R. Nikmehr, M. J. |
| author_sort |
Jahanbakhsh, N. |
| title |
On the inclusion ideal graph of a poset |
| title_short |
On the inclusion ideal graph of a poset |
| title_full |
On the inclusion ideal graph of a poset |
| title_fullStr |
On the inclusion ideal graph of a poset |
| title_full_unstemmed |
On the inclusion ideal graph of a poset |
| title_sort |
on the inclusion ideal graph of a poset |
| description |
Let \((P, \leq)\) be an atomic partially ordered set (poset, briefly) with a minimum element \(0\) and \(\mathcal{I}(P)\) the set of nontrivial ideals of \( P \). The inclusion ideal graph of \(P\), denoted by \(\Omega(P)\), is an undirected and simple graph with the vertex set \(\mathcal{I}(P)\) and two distinct vertices \(I, J \in \mathcal{I}(P) \) are adjacent in \(\Omega(P)\) if and only if \( I \subset J \) or \( J \subset I \). We study some connections between the graph theoretic properties of this graph and some algebraic properties of a poset. We prove that \(\Omega(P)\) is not connected if and only if \( P = \{0, a_1, a_2 \}\), where \(a_1, a_2\) are two atoms. Moreover, it is shown that if \( \Omega(P) \) is connected, then \( \operatorname{diam}(\Omega(P))\leq 3 \). Also, we show that if \( \Omega(P) \) contains a cycle, then \( \operatorname{girth}(\Omega(P)) \in \{3,6\}\). Furthermore, all posets based on their diameters and girths of inclusion ideal graphs are characterized. Among other results, all posets whose inclusion ideal graphs are path, cycle and star are characterized. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2019 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/261 |
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AT jahanbakhshn ontheinclusionidealgraphofaposet AT nikandishr ontheinclusionidealgraphofaposet AT nikmehrmj ontheinclusionidealgraphofaposet |
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2025-12-02T15:26:31Z |
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2025-12-02T15:26:31Z |
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