Line intersection graphs of ideals of a poset

Subject Areas : Algebra

Soheila Khojasteh ¹

1 - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

Received: 2022-11-07 Accepted : 2023-04-18 Published : 2024-09-26

Keywords: گراف خطی, مجموعه جزئاً مرتب, مکمل گراف اشتراکی, گراف اشتراکی,

Abstract :

Let (P,≤) be an atomic poset with the least element 0. The intersection graph of ideals of P denoted by G(P), is defined to be a graph whose vertices are all non-trivial ideals of P and two distinct vertices I and J are adjacent if and only if I∩J≠{0}. The complement of G(P) is denoted by Γ(P) Also, the .line graph of a graph G is denoted by L(G), is a graph whose vertex set is equal to the edge set of G, and two distinct vertices of L(G) are adjacent if and only if their corresponding edges are incident in G In this paper, we .determine all posets P for which G(P) or Γ(P) is a line graph. We prove that Γ(P) is a line graph if and only if |Atom(P)|=1 or Atom(P)={a_1,a_2 } such that |{a_1 }^u\〖\{a_2}〗^u |, |{a_2 }^u\\{a_1 }^u |≤2 or |Atom(P)|=3 with P=Atom(P)∪{0} or |Atom(P)|=3 and there exists natural number say n such that P= Atom(P)∪{0}∪{b_1,…,b_n }, and for every a∈Atom(P),{b_1,…,b_n }⊆{a}^u.

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