On the inclusion ideal graph of a poset

N. Jahanbakhsh, R. Nikandish, M. J. Nikmehr


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.


poset, inclusion ideal graph, diameter, girth, connectivity

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