Geodesic
On the inclusion ideal graph of a poset
Algebra and discrete mathematics, Tome 27 (2019) no. 2, pp. 269-279.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: poset, inclusion ideal graph, diameter, girth, connectivity. 
@article{ADM_2019_27_2_a8,
     author = {N. Jahanbakhsh and R. Nikandish and M. J. Nikmehr},
     title = {On the inclusion ideal graph of a poset},
     journal = {Algebra and discrete mathematics},
     pages = {269--279},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2019_27_2_a8/}
}
TY  - JOUR
AU  - N. Jahanbakhsh
AU  - R. Nikandish
AU  - M. J. Nikmehr
TI  - On the inclusion ideal graph of a poset
JO  - Algebra and discrete mathematics
PY  - 2019
SP  - 269
EP  - 279
VL  - 27
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2019_27_2_a8/
LA  - en
ID  - ADM_2019_27_2_a8
ER  - 
%0 Journal Article
%A N. Jahanbakhsh
%A R. Nikandish
%A M. J. Nikmehr
%T On the inclusion ideal graph of a poset
%J Algebra and discrete mathematics
%D 2019
%P 269-279
%V 27
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2019_27_2_a8/
%G en
%F ADM_2019_27_2_a8
N. Jahanbakhsh; R. Nikandish; M. J. Nikmehr. On the inclusion ideal graph of a poset. Algebra and discrete mathematics, Tome 27 (2019) no. 2, pp. 269-279. http://geodesic.mathdoc.fr/item/ADM_2019_27_2_a8/

[1] S. Akbari, M. Habibi, A. Majidinia, R. Manaviyat, “The inclusion ideal graph of rings”, Comm. Algebra, 43 (2015), 2457–2465 | DOI | Zbl

[2] J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Elsevier, New York, 1976 | Zbl

[3] Y. Civan, “Upper maximal graphs of posets”, Order, 30 (2013 677–688) | DOI | Zbl

[4] R. Halaš, M. Jukl, “On Beck's coloring of posets”, Discrete Math., 309 (2009), 4584–4589 | DOI | Zbl

[5] V. Joshi, “Zero divisor graph of a poset with respect to an ideal”, Order, 29 (2012), 499–506 | DOI | Zbl

[6] J. D. LaGrange, K. A. Roy, “Poset graphs and the lattice of graph annihilators”, Discrete Math., 313 (2013), 1053–1062 | DOI | Zbl

[7] S. K. Nimbhorkar, M. P. Wasadikar, L. DeMeyer, “Coloring of semilattices”, Ars Combin, 12 (2007), 97–104

[8] S. Roman, Lattices and Ordered Sets, Springer, New York, 2008 | Zbl

[9] S. Rudeanu, Sets and Ordered Structures, University of Bucharest, Romania, 2012

[10] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River, 2001