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The complement of the intersection graph of ideals of a poset.
- Source :
-
Journal of Algebra & Its Applications . Nov2023, Vol. 22 Issue 11, p1-13. 13p. - Publication Year :
- 2023
-
Abstract
- Let (P , ≤) be an atomic poset with the least element 0. The complement of the intersection graph of ideals of P , denoted by Γ (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 }. In this paper, we consider the complement of the intersection graph of ideals of a poset. We prove that Γ (P) is totally disconnected or diam (Γ (P) \) ∈ { 1 , 2 , 3 } , where is the set of all isolated vertices of Γ (P). We show that g r (Γ (P)) ∈ { 3 , 4 , ∞ }. Also, we characterize all posets whose complement of the intersection graph is forest, unicyclic or complete r -partite graph. Among other results, we prove that Γ (P) is weakly perfect; and it is perfect if and only if | Atom (P) | ≤ 4. Finally, we show that Γ (P) is class 1 , where P = Atom (P) ∪ { 0 }. [ABSTRACT FROM AUTHOR]
- Subjects :
- *INTERSECTION graph theory
*PARTIALLY ordered sets
Subjects
Details
- Language :
- English
- ISSN :
- 02194988
- Volume :
- 22
- Issue :
- 11
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra & Its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 173273206
- Full Text :
- https://doi.org/10.1142/S0219498823502365