On the inclusion ideal graph of a poset

Abstract

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.