On the inclusion ideal graph of a poset
Let (P,≤) be an atomic partially ordered set (poset, briefly) with a minimum element 0 and 𝕿(P) the set of nontrivial ideals of P. The inclusion ideal graph of P, denoted by Ω(P), is an undirected and simple graph with the vertex set 𝕿(P) and two distinct vertices I, J ∈ 𝕿(P) are adjacent in Ω(P) if...
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| Date: | 2019 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2019
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| Series: | Algebra and Discrete Mathematics |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/188437 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the inclusion ideal graph of a poset / N. Jahanbakhsh, R. Nikandish, M.J. Nikmehr // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 269–279. — Бібліогр.: 10 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | Let (P,≤) be an atomic partially ordered set (poset, briefly) with a minimum element 0 and 𝕿(P) the set of nontrivial ideals of P. The inclusion ideal graph of P, denoted by Ω(P), is an undirected and simple graph with the vertex set 𝕿(P) and two distinct vertices I, J ∈ 𝕿(P) are adjacent in Ω(P) if and only if I ⊂ J or J ⊂ I. We study some connections between the graph theoretic properties of this graph and some algebraic properties of a poset. We prove that Ω(P) is not connected if and only if P = {0, a1, a2}, where a1, a2 are two atoms. Moreover, it is shown that if Ω(P) is connected, then diam(Ω(P)) ≤ 3. Also, we show that if Ω(P) contains a cycle, then girth(Ω(P)) ∈ {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. |
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